2024 - 2025
Organized by Laurent Bartholdi, Anna Erschler and Cyril Houdayer
Partially supported by ERC Advanced Grant 101097307 (P.I.:Laurent Bartholdi)
and ERC Advanvced Grant ERC 101141693 (P.I.: Cyril Houdayer)
January 15 (Wednesday)
ENS, Salle W
14.00 - 14.45 Bram Petri
15.15 - 16.00 Federica Gavazzi
16.15 - 17.00 Amandine Escalier
Group theory day, November 22, 2024
IAS Princeton (Simonyi Hall 101)
and streamed via zoom;
https://theias.zoom.us/j/83472608941?pwd=FMNH20u3wzJEB1xxtIxzu1BtEwyrah.1
Morning session, chaired by Kasia Jankiewicz
09.00 - 09.40 Catherine Pfaff (IAS)
09.45- 10.25 Kunal Chawla (PU)
11.00-11.40. Josh Frisch (UCSD)
Afternoon session, chaired by Catherine Pfaff
13.15 - 13.55 Kasia Jankiewicz (IAS)
14.00 - 14.40 Doron Puder (IAS)
15.00-15.30. tea break in the Fuld Hall
15.30 - 16.15 Sam Taylor (Temple University) (CANCELLED)
Catherine Pfaff
Title: Train track automata for outer automorphisms of free groups and geodesics in outer space
Abstract: The outer automorphism group of the free group Out(F_r) acts as the isometry group on the deformation space of weighted graphs, Culler-Vogtmann Outer space CV_r. The train track theory of Bestvina-Feighn-Handel bridges studying topological representatives of the group elements and geodesics in this space it acts on. We use the asymptotic conjugacy class invariant of the Handel-Mosher ideal Whitehead graph to “stratify” the space of geodesics, and the dynamically minimal “fully irreducible” outer automorphisms, into train track automata for different ideal Whitehead graphs. We then also contextualize this work in the broader program of understanding the geodesic flow. While the flow in the closed hyperbolic manifold and Teichmuller space settings is ergodic, it is unclear whether graphs live in such a nice setting. We explain some of our indicators of certain properties the flow may have. Some results presented are joint with some combinations of Y. Algom-Kfir, D. Gagnier, I. Kapovich, J. Maher, L. Mosher, and S.J. Taylor.
Kunal Chawla
Title: Detecting Stationary Measure Classes
Abstract: A random walk on a hyperbolic group G will converge almost surely to the boundary, defining a stationary measure at infinity. When G is a cocompact fuchsian group, this boundary is the circle, and when the random walk is finitely supported it is unknown whether this measure can ever be absolutely continuous with respect to the Lebesgue measure. We provide a simple and geometric family of statistics which determine whether two stationary measures are in the same measure class.
Josh Frisch
Title: Computing Poisson Boundaries without moment conditions
Abstract: The Poisson boundary is a measure theoretic object which classifies both the space of possible asymptotic events of a random walk and the space of bounded harmonic functions. For most identifications so far a moment condition has proved crucial. I will talk about joint works with Kunal Chawla, Behrang Forghani, Giullio Tiozzo and with Eduardo Silva where we show that, for the most part, these moment conditions are unnecessary. In particular I will talk about the classes of acylindrically hyperbolic groups and wreath products.
Kasia Jankiewicz
Title: Profinite properties of Artin groups
Abstract: Artin groups are a family of groups generalizing braid groups, and are given by simple looking group presentations. In this talk, I will discuss profinite properties of Artin groups, such as residual finiteness, and the connection between the group and its profinite completion. This will include joint work with Kevin Schreve.
Doron Puder
Title: Detecting primitivity and free-splittings in the free group algebra
Abstract: The free group algebra appears often as a major tool in dealing with problems about free groups. It is also a lovely object for its own sake, featuring many analogies with the free group: for example, it satisfies an analog of the Nielsen-Schreier theorem. Yet, while many algorithms exist for different problems in free groups, there are very few algorithms for basic problems in the free group algebra.
I plan to define the free group algebra, try to convince you it is a nice and interesting creature, and introduce a new algorithm for detecting whether a given element is "primitive" in a given ideal (what Whitehead's algorithm achieves in free groups).
Based on joint work with Matan Seidel and Danielle Ernst-West.
Sam Taylor (The talk is cancelled)
Title: Simultaneous universal circles
Abstract: Let M be a connected, oriented, closed, irreducible, and atoroidal 3-manifold. Two structures of central importance in 3-manifold theory are taut foliations and pseudo-Anosov flows. Each structure determines an action of the fundamental group of M on a circle, both of which are often referred to as a universal circle for the structure. In this talk, I’ll report on work-in-progress with Landry and Minsky that describes a precise relation between these: the circle action associated to a pseudo-Anosov flow on M is a universal circle for any taut foliation on M that is (almost) transverse to the flow
Group theory seminar, ENS, Paris.
2024 - 2025
Organized by Laurent Bartholdi and Anna Erschler
Partially supported by ERC Advanced Grant 101097307 (P.I.:Laurent Bartholdi).
2023-2024
Organized by Andrei Alpeev, Laurent Bartholdi, Anna Erschler and Panagiotis Tselekidis
Partially supported by ERC Advanced Grant 101097307 (P.I.:Laurent Bartholdi).
May 29 (Wednesday)
An afternoon on random walks on groups.
14.00 - 14.45 Giulio Tiozzo (Toronto), "Roots of Alexander polynomials of random positive braids"
15.00 - 15.45 Amaury Freslon (Orsay) "How to (badly) shuffle cards?"
16.15 - 17.00 Charles Bordenave (Marseille) "Strong convergence of matrix algebras and applications to random walks".
Giulio Tiozzo "Roots of Alexander polynomials of random positive braids".
As originally observed experimentally by Dehornoy, roots of Alexander polynomials of random knots display interesting patterns. In this work, joint with N. Dunfield, we prove several results on the distribution of such roots in the complex plane, and discuss further conjectures that originate from them.
Using the Burau representation, this corresponds to studying random walks on the group SL(2, C[t]) of 2-by-2 matrices with polynomial coefficients. We compute a sharp lower bound on the probability that such roots lie on the unit circle, and prove a related central limit theorem. We also show there is a large root-free region near the origin.We introduce the notion of a Lyapunov exponent for the Burau representation, in the spirit of Deroin-Dujardin, and a corresponding bifurcation measure, which we prove to be the limiting measure for the distribution of roots on a region of parameter space.
Amaury Freslon "How to (badly) shuffle cards?"
Card shuffling can be modelled by random walks on permutation groups, and the first example which was studied in depth is the one given by random transpositions. In that case, Diaconis and Shahshahani proved that the corresponding Markov chain exhibits a so-called cut-off phenomenon. Moreover, Teyssier recently computed the corresponding cut-off profile, which is remarkably simple. I will explain how one can similarly define a random walk on the "quantum permutation groups", a Hopf algebra which somehow contains the usual permutation groups. I will then report on a joint work with Teyssier and Wang where we prove the cut-off phenomenon for that process and compute the cut-off profile.
Charles Bordenave "Strong convergence of matrix algebras and applications to random walks".
We will present results on the convergence of the operator norm of random matrices of large dimension. Our random matrices are build by taking tensor products of deterministic matrices and independent Haar distributed unitary matrices or independent random permutation matrices. This class of random matrices allows for example to consider random Schreier graphs of the modular group or of Cartesian products of free groups. We will explain how these convergence results can be used to prove sharp mixing time estimates on random walks. The talk will be notably based on joint works with Benoit Collins and Hubert Lacoin.
April 16 (Tuesday).
14.00 - 14.45 Andrei Jaikin-Zapirain ( UAM Madrid), "Compressed subgroups in free groups are inert".
15.00 - 15.45 Timothée Marquis (UCLovain), , "Amalgams of rational unipotent groups and residual nilpotence".
16.15 - 17.00 Olga Kharlampovich (CUNY Graduate Center and Hunter College) "Quantification of separability of cubically convex-cocompact subgroups
of RAAGs via representations".
Andrei Jaikin-Zapirain "Compressed subgroups in free groups are inert".
Let F be a free group. A finitely generated subgroup H is called compressed in F if it is not contained in a subgroup of F of smaller rank than H, and it is called inert in F if H ∩ U is compressed in U for any subgroup U of F. In my talk, I will show that compressed subgroups are also inert. The solves a conjecture of Dicks and Ventura from 1996.
Timothée Marquis , "Amalgams of rational unipotent groups and residual nilpotence".
Given a group property (P), a group G is called residually (P) if every nontrivial element of G has a nontrivial image in some quotient of G that satisfies (P). The study of residual properties of graphs of groups has a long and rich history, originating from Magnus’ theorem that free groups are residually torsionfree nilpotent. In this talk, I will start by reviewing a few key results of this history, before presenting an intriguing phenomenon concerning the residual nilpotence of certain amalgams of rational unipotent groups. Joint work with Pierre-Emmanuel Caprace.
Olga Kharlampovich, "Quantification of separability of cubically convex-cocompact subgroups
of RAAGs via representations".
We answer the question asked by Louder, McReinolds and Patel and prove the following statement. Let L be a RAAG, H a
cubically convex-cocompact subgroup of L, then there is a finite dimensional representation of L that separates the subgroup H
in the induced Zariski topology. As a corollary, we establish a polynomial upper bound on the size of the quotients used to separate H
in L. This implies the same statement for a virtually special group L and, in particular, a fundamental group of a hyperbolic 3-manifold.
For any finitely generated subgroup H of a limit group L we prove the same results and, in addition, show that there exists a finite-index
subgroup K containing H, such that K is a subgroup of a group obtained from H by a series of extensions of centralizers and free products
with infinite cyclic group. If H is non-abelian, the K is fully residually H. A corollary is that a hyperbolic limit group satisfies
the Geometric Hanna Neumann conjecture. These are joint results with K. Brown and A. Vdovina.
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An afternoon on Random Walks on Groups
to the memory of Anatoly Moiseevich Vershik
March 12 (Tuesday)
14.00 - 14.45 Cyril Houdayer (ENS Paris) "The noncommutative factor theorem for lattices in product groups".
15.00 - 15.45 Kunal Chawla (Princeton) "The Poisson boundary of hyperbolic groups without moment conditions".
16.15 - 17.00 Sara Brofferio (Université Paris-Est Créteil Val-de-Marne), "Uniqueness of invariant measures for random homeomorphisms of the real line".
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Cyril Houdayer "The noncommutative factor theorem for lattices in product groups ".
In this talk, I will present a noncommutative analogue of Bader-Shalom factor theorem for lattices with dense projections in product groups. Combining with previous works, this result provides a noncommutative analogue of Margulis factor theorem for all irreducible lattices in higher rank semisimple algebraic groups. Namely, we give a complete description of all intermediate von Neumann subalgebras sitting between the group von Neumann algebra associated with the lattice and the group measure space von Neumann algebra associated with the action of the lattice on the Furstenberg-Poisson boundary. This is joint work with Rémi Boutonnet.
Kunal Chawla, "The Poisson boundary of hyperbolic groups without moment conditions".
Forty years ago, Kaimanovich and Vershik introduced fundamental entropic techniques in the study of Poisson boundaries. A conditional entropic criterion of Kaimanovich was often applied in the form of 'strip' and 'ray' approximation criteria. In this talk, we present a new method, the 'pin down approximation criterion' which, combined with new geometric and probabilistic techniques, allows us to identify the Poisson boundary for hyperbolic groups with finite entropy measures, without any moment assumptions. Our method provides the first examples of a Non-Choquet Deny group where the Poisson boundary can be identified for all finite entropy measures. No prior knowledge of hyperbolic groups will be assumed. This is joint work with Behrang Forghani, Joshua Frisch, and Giulio Tiozzo.
Sara Brofferio "Uniqueness of invariant measures for random homeomorphisms of the real line".
A stochastic dynamical systems is a Markov process defined recursively by X_n=\Psi_n(X_{n-1})=\Psi_n\cdots \Psi_1(X_0) where
\Psi_n i.i.d. random continuous transformations on a given space M. X_n can be seen as the process obtained by the action of the random walk \Psi_n\cdots \Psi_1 on a (semi)-group Gamma acting on some nice metric space M.
In this talk we will focus on stochastic dynamical systems induced by a random walk on the group of homeomorphisms of R. I will present the results of joint work with D. Buraczewski and T. Szarek, in which we establish conditions (relatively optimal) that guarantee that the system admits a unique invariant measure (possibly of infinite mass).
January 31 (wednesday)
14.00 - 14.45 Valérie Berthé (Paris VII), "Dendric subshifts and groups"
15.00 - 15.45 Nguyen-Bac Dang, (Orsay) , "Variation of the Hausdorff dimension of limits set and degenerating
Schottky groups"
16.15 - 17.00 Bruno Duchesne (Orsay), "The isometry group of the infinite dimensional hyperbolic space ".
Valérie Berthé, "Dendric subshifts and groups"
We discuss a family of symbolic dynamical systems that have remarkable group properties, the family of dendric words. This family includes numerous classical families of symbolic dynamical systems, among others codings of interval exchanges. Their return words form positive basis of the free group. We discuss their dimension groups, which are complete invariants of strong orbit equivalence, and applications to skew products based on finite groups.
Nguyen Bac Dang, "Variation of the Hausdorff dimension of limits set and degenerating
Schottky groups"
In this talk, based on a joint work with Vlerë Mehmeti, I will explain how one can use some techniques in non-Archimedean geometry to
study families of degenerating complex Schottky groups. More precisely, each Schottky group comes with a fractal set, obtained as a limit of an
orbit, called the limit set. We show that under specific conditions, one can can obtain an asymptotic formula for the Hausdorff dimension of
the limit set. If time permits, I will present how certain functions, called Poincare series have very special behavior when one works over non-Archimedean fields.
Bruno Duchesne "The isometry group of the infinite dimensional hyperbolic space ". After motivating the study of the infinite dimensional hyperbolic space, we will focus on its isometry group as topological group. It is a Polish group which is topologically simple with the automatic continuity property. Moreover, we understand quite well its topological dynamics since its universal minimal flow and its Furstenberg boundary can be explicitly described.
December 13 (wednesday, unusual day)
14.00-14.45 Reem Yassawi (London) "Tame or wild Toeplitz shifts"
15.00-15.45 Todor Tsankov (Lyon) "Gleason complete flows of locally compact groups"
16.15-17.00 Johannes Kellendonk (Lyon) "Which algebraic components of the Ellis semigroup of a non-tame dynamical system are especially big?"
Reem Yassawi "Tame or wild Toeplitz shifts"
The Ellis semigroup E(X, T) of a topological dynamical system is defined to be the compactification of the action T in the topology of pointwise convergence on the space of all functions X^X. Tameness is a concept whose roots date back to Rosenthal’s ℓ^1 embedding theorem, which says that if a sequence in ℓ^1 does not have a weakly Cauchy subsequence, then it must be the sequence of unit vectors in ℓ^1. Köhler linked the concept of tameness to the Ellis semigroup. A system is tame if its Ellis semigroup has size at most the continuum. Non-tame systems are very far from tame, as they must contain a copy of βℕ, the Stone-Čech compactification of ℕ.
Since then, the dynamics community has investigated the question of which systems are tame. In this talk I will give a brief exposition of these results, and talk about work where we study tameness, or otherwise, of Toeplitz shifts, emphasizing the connection between this work and automata. This is joint work with Gabriel Fuhrmann and Johannes Kellendonk.
Todor Tsankov "Gleason complete flows of locally compact groups"
The notion of an irreducible extension of a flow generalizes the one of an almost one-to-one extension (injective on a dense G_delta set)
and coincides with the one of a highly proximal extension for minimal flows. The existence of maximal such extensions was proved by
Auslander and Glasner in the 70s for minimal flows using an abstract argument, and a concrete construction using near-ultrafilters was
recently given by Zucker for arbitrary flows. When the acting group is discrete, the universal irreducible extension is nothing but the Stone
space of the Boolean algebra of the regular open sets of the space, already considered by Gleason. We give yet another construction of the
universal irreducible extension for arbitrary topological groups and prove that for such extensions (which we call Gleason complete) of
a flow of a locally compact group G, the stabilizer map x -> G_x is continuous (for general flows, this map is only semi-continuous). This
is a common generalization of a theorem of Frolík that the set of fixed points of a homeomorphism of a compact, extremally disconnected
space is open and a theorem of Veech that the action of a locally compact group on its greatest ambit is free. The theorem implies, in particular, that if the action of a locally compact group on its Furstenberg boundary is essentially free, then it is free. This is joint work with Adrien Le Boudec.
Johannes Kellendonk "Which algebraic components of the Ellis semigroup of a non-tame dynamical system are especially big?"
The Ellis semigroup E of a group acting by homeomorphisms on a compact space is its compactification in the topology of point wise convergence. It has a lot of interesting structures: its topology, the topological properties of its elements, and its algebraic structure. One property which has incited of lot of interest in recent years is tameness. In can be characterised in various different ways, but for our talk the quickest way is to say is that E is tame if its cardinality is at most that of the continuum. So non-tame Ellis semigroups are especially big. We are interested in how this relates to the algebraic structure of the Ellis semigroup. For instance, when is the kernel of the Ellis semigroup especially big? A recent result shows that, if the set of idempotents of a minimal right ideal of the Ellis-semigroup of a minimal system is especially big, then the system cannot be a PI-flow. It is also known that for minimal actions of groups which do not carry an invariant measure, tameness implies that the system is almost automorphic. In both cases the converse is not true (for almost automorphic non-tame systems see in particular the talk by Reem Yassawi). We will show here that for minimal abelian group actions which are not almost automorphic and whose set of singular points satisfies a condition which will be specified, the kernel of the Ellis semigroup is especially big, and here it is in particular the Rees structure group which is especially big.
Nomember 15 (wednesday, unusual day)
14.00-14.45 Simon André (Paris VI) "Sharply 2-transitive infinite finitely generated simple groups".
15.00-15.45 Ruiwen Dong (Saarland University) "Decision problems in sub-semigroups of metabelian groups".
16.00-16.45 Emmanuel Rauzy (Munich University) "Groups with presentations in EDT0L".
Simon André "Sharply 2-transitive infinite finitely generated simple groups".
A group G is said to be sharply 2-transitive if it has an action on a set X with at least 2 elements such that, for all pairs (x, x') and (y, y') of distinct elements in X, there exists a unique element g in G such that g(x, x') = (y, y'). For example, the affine group AGL(1, K) over a field K is sharply 2-transitive (for its natural action on K), and quite surprisingly, the following question remained open for a long time: does there exist a sharply 2-transitive group that is not isomorphic to some AGL(1, K)? A few years ago, Rips, Segev, and Tent constructed the first example of a sharply 2-transitive group that is not affine. In my talk, I will explain that we can go further and construct various sharply 2-transitive groups that are radically different from affine groups. These results were obtained in collaboration with Marco Amelio, Vincent Guirardel, and Katrin Tent.
Ruiwen Dong "Decision problems in sub-semigroups of metabelian groups".
Algorithmic problems in metabelian groups have been studied as early as the 1950s since the work of Hall. In the 1970s Romanovskii proved decidability of the Group Membership problem (given the generators of a subgroup and a target element, decide whether the target element is in the subgroup) in metabelian groups. However, Semigroup Membership (same as Group Membership, but with sub-semigroups) has been shown to be undecidable in several instances of metabelian groups using embeddings of either the Hilbert's tenth problem or two-counter automata.
In this talk we consider two "intermediate" decision problems: the Identity Problem (deciding if a sub-semigroup contains the neutral element) and the Group Problem (deciding if a sub-semigroup is a group). We reduce them to solving linear equations over the polynomial semiring N[X] and show decidability using an extension of a local-global principle by Einsiedler (2003).
Emmanuel Rauzy "Groups with presentations in EDT0L"
There are numerous connections between group theory and language theory, which for the most part stem from the fact that elements of a finitely generated group are commonly represented by words on the generators.
Out of these connections, one of the least studied ones is the notion of a group that admits a presentation in a given class of languages. Indeed, while the notions of finite presentations and of recursive presentations are commonly invoked, finite languages and recursive languages correspond to the two extremes of the Chomsky hierarchy -there is much in between!
We show that the groups that admit an L-presentation, a notion introduced by Bartholdi in 2000, correspond exactly to those that admit EDT0L presentations, a class of language which has been the focus of much attention in group theory following work of Ciobanu and Elder.
We present a uniform proof for the fact that one can compute finite, nilpotent, metabelian and free quotients of a group described by an EDT0L presentation, extending results of Bartholdi, Eick, Hartung. This proof relies on subgroup functors that satisfy some Noetherianity conditions.
Finally, we explain how these results allow us to produce examples of recursively presented groups that do not admit EDT0L presentations.
This is joint work with Laurent Bartholdi and Leon Pernak.
Tuesday October 10, room W, ENS
14.00-14.45 Timothée Bénard (University of Warwick), "Limit theorems on nilpotent Lie groups".
I will present limit theorems for random walks on nilpotent Lie groups, obtained in a recent work with Emmanuel Breuillard. Most works on the topic assumed the step distribution of the walk to be centered in the abelianization of the group. Our main contribution is to authorize non-centered step distributions. In this case, new phenomena emerge: the large-scale geometry of the walk depends on the increment average, and the limit measure in the central limit theorem may not have full support in the ambient group.
15.00-15.45 Panagiotis Tselekidis (ENS Paris), "Asymptotic dimension of finitely generated groups".
Asymptotic dimension introduced by Gromov as an invariant of finitely generated groups.It can be shown that if two metric spaces are quasi-isometric then they have the same asymptotic dimension. In 1998, the asymptotic dimension achieved particular prominence in geometric group theory after a paper of Guoliang Yu, which proved the Novikov conjecture for groups with finite asymptotic dimension. Unfortunately, not all finitely generated groups have finite asymptotic dimension. In this talk, we will introduce some basic tools to compute the asymptotic dimension of groups. We will also find upper bounds for the asymptotic dimension of a few well-known classes of finitely generated groups, and if time permits, we will see why one-relator groups have asymptotic dimension at most two.
16.15 - 17.00 Andrey Alpeev (ENS Paris), "Invariant random order extension and amenability".
Classical order extension principle states that any partial order on a set could be extended to a linear order. An invariant random order is a measure on the space of all partial order on a given (countable) group that is invariant under the natural shift-action. Is it always possible to extend an invariant random partial order to an invariant random linear order? The answer is affirmative in the case of an amenable group. First negative result was obtained by Y. Glasner, Lin, and Meyerovitch , who proved that there is an invariant order on SL_3(Z) that cannot be extended to invariant random linear order. I will show that their example could be transferred to all non-amenable groups, thus proving that the invariant random order extension property is equivalent to amenability.
zoom link:
https://cs-uni-saarland-de.zoom.us/j/82496354750
Meeting ID: 824 9635 4750
passoword: the cardinality of the smallest non-abelian group.
video of the last seminars
https://www.youtube.com/@ENSGroupTheorySeminar/playlists
2022-2023
Organized by Anna Erschler, Josh Frisch, Ivan Mitrofanov and Rachel Skipper
Tuesday May 16
14.00-14.45 Jingyin Huang (Ohio State University)
15.00-15.45 Corentin Bodard (University of Geneva)
16.00-16.45 Lopez Neumann Antonio (École polytechnique)
Jingyin Huang "Labeled four wheels and the K(pi,1) problem for reflection arrangement complements"
The K(π,1)-conjecture for reflection arrangement complements, due to Arnold, Brieskorn, Pham, and Thom, predicts that certain complexified hyperplane complements associated to infinite reflection groups are Eilenberg MacLane spaces. We establish a close connection between a very simple property in metric graph theory about 4-cycles and the K(π,1)-conjecture, via elements of non-positively curvature geometry. We also propose a new approach for studying the K(π,1)-conjecture. As a consequence, we deduce a large number of new cases of Artin groups which satisfies the K(π,1)-conjecture.
Corentin Bodard "Intermediate geodesic growth in virtually nilpotent groups".
Geodesic growth counts the number of geodesics of length n in the Cayley graph of a group G (with a generating set S). As soon as the group has exponential volume growth, the geodesic growth is also exponential. Finding pairs (G,S) with polynomial geodesic growth is trickier but, by now, several constructions are known. At last, a question that has been around and open since the 90's is the existence of a pair (G,S) with intermediate geodesic growth. Most of the efforts to construct such an example have been centered on groups of intermediate volume growth, without success. Perhaps surprisingly, we show that intermediate geodesic growth is possible in the realm of virtually nilpotent groups. In this talk, I will introduce our main example, a virtually 3-step nilpotent group, and a geometric model for it. I will explain the main ideas necessary for both the upper and lower bounds on the number of geodesics, and compare them with previous arguments of Shapiro, Bridson-Burillo-Elder-Sunic and Bishop-Elder.
Lopez Neumann Antonio "Vanishing of the second L^p-cohomology group for most semisimple groups of rank at least 3".
L^p-cohomology is a quasi-isometry invariant popularized by Gromov. For semisimple groups over local fields, he predicts a classical behaviour (i.e. vanishing for every p>1) in degrees below the rank. It was shown by Pansu that this is true in degree 1 for all higher rank real semisimple groups. We show vanishing of L^p-cohomology in degree 2 for most semisimple groups of rank at least 3, for all p > 1 in the non-Archimedean case and for large p in the real case. In this talk we will review previous contributions on this question and sketch a proof of this result.
Tuesday April 11
14.00 -14.45 Alex Eskin (University of Chicago)
15.00- 15.45 Alex Gamburd (Graduate Center, CUNY) (cancelled!)
15.00-15.45 Vadim Kaimanovich (University of Ottawa)
Alex Eskin "On a theorem of Furstenberg".
A deep result of Furstenberg from 1967 states that if Gamma is a lattice in a semisimple Lie group G, then there exists a measure onGamma with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary is absolutely continuous. I will discuss some of the history of this result and some recent generalizations.
Alex Gamburd "Arithmetic and Dynamics on Varieties of Markoff Type".
The Markoff equation $x^2+y^2+z^2=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss recent progress towards establishing forms of strong approximation on varieties of Markoff type, as well as ensuing implications, diophantine and dynamical. (Joint work with J. Bourgain and P. Sarnak)
Vadim Kaimanovich, "Limit distributions of branching random walks"
Various setups lead to a consideration of sequences (families) of probability measures on a given group and of their asymptotic behaviour.
Arguably, the most natural example is given by the sequence of convolution powers of a fixed measure. In probabilistic terms this is the sequence of one-dimensional distributions of the associated random walk on the group. This interpretation then allows one to consider the resulting (random) sample paths (or the associated random sequences of delta measures) as well.
A branching random walk can be considered as an intermediate between these two extremes. Roughly speaking, it arises from a combination of two classical Markov chains: an ordinary random walk and a Galton – Watson branching process. The state space of a branching random walk consists of inite populations (i.e., of finite integer valued occupation measures), and their members are independently subject to the same group invariant random fission-displacement mechanism. By varying the branching ratio one obtains a continuous interpolation between random sample paths of ordinary random walks (when there is no branching at all, and the branching ratio is 1) and deterministic sequences of convolution powers (in the limit when the branching ratio tends to infinity).
There are numerous questions that can be formulated in the framework of this approach. In particular, passing to the empirical distributions of random populations by normalization of the occupation measures one can ask about the weak limits of these empirical distributions in the presence of an appropriate group compactification.
The talk (based on a joint work with Wolfgang Woess) will be devoted to a discussion of this problem. No prior knowledge of the theory of branchingprocesses is required.
Tuesday March 14
14.00-14.45 Emmanuel Breuillard (Oxford)
15.00 -15.45 Pierre Py (Strasbourg)
16.00 - 16.45 Olga Kharlampovich (Graduate Center and Hunter College CUNY)
Emmanuel Breuillard "Random character varieties". We study the representation and character varieties of random finitely presented groups with values in a complex semisimple Lie group. We compute their dimension and number of irreducible components. For example we show that for all but exponentially few pairs of words (w_1,w_2) of length at most n, all homomorphic images of the finitely presented group <a,b| w_1=w_2=1> in GL(d,C) have virtually solvable image, and that random one-relator groups <a,b | w> have many rigid Zariski-dense representations. The proofs are conditional on GRH and use new results regarding expander Cayley graphs of finite simple groups of Lie type as a key ingredient. (Joint work with O. Becker and P. Varju)
Pierre Py "Subgroups of hyperbolic groups, finiteness properties and complex hyperbolic lattices". Following C.T.C. Wall, we say that a group is of type F_n if it admits a classifying space which is a CW-complex with finite n-skeleton. For n = 2, one recovers the notion of being finitely presented. We prove that in a cocompact arithmetic lattice in the group PU(m,1), with positive first Betti number, deep enough finite index subgroups admit plenty of homomorphisms to Z with kernel of type F_{m−1} but not of type F_m. This provides many non-hyperbolic finitely presented subgroups of hyperbolic groups and answers an old question of Brady. This is based on a joint work with C. Llosa Isenrich.
Olga Kharlampovich "Equations and first-order sentences in random groups".
We prove that a random group, in Gromov's density model with d<1/16 (such a random group is a small cancellation group) satisfies with overwhelming probability a universal-existential first-order sentence sigma (in the language of groups) if and only if sigma is true in a nonabelian free group. This is based on our result that solutionsof a system of equations in such a random group with overwhelming probability are images of solutions in a free group. We also discuss problems with higher densities that come from differences between hyperbolic and small cancellation groups. These are joint results with R. Sklinos.
Wendesday February 22 (unusual day)
14.00-14.45 Yves Benoist (Orsay) "Harmonic functions on the Heisenberg group".
15.00 -15.45 Nora Szakacs (Manchester) "Inverse semigroups as metric spaces, and their uniform Roe algebras".
16.00 - 16.45 Richard Aoun (Marne-la-Vallée) "Concentration inequalities for random walks on hyperbolic spaces, and applications".
Yves Benoist "Harmonic functions on the Heisenberg group".
A harmonic function on a group G is a function which is equal to the average of its translates. I will first recall classical results of Choquet-Deny, Margulis and Ancona for abelian, nilpotent and hyperbolic groups. Then I will describe all the positive harmonic functions on the Heisenberg group.
Nora Szakacs "Inverse semigroups as metric spaces, and their uniform Roe algebras".
Given any quasi-countable, in particular any countable inverse semigroup S, we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S. This allows us to unambiguously define the uniform Roe algebra of S, which is a C*-algebra capturing the large scale geometry of the space. Using this setting, we study those inverse semigroups with asymptotic dimension 0. Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups, and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally $\mathcal L$-finite, and equivalently sparse as a metric space.This work is joint with Yeong Chyuan Chung and Diego Martínez
Richard Aoun "Concentration inequalities for random walks on hyperbolic spaces, and applications".
Let (X,d) be a Gromov hyperbolic space on which a group G acts by isometries and non elementarily. An i.i.d random walk R_n=X_1...X_n (driven by a probability measure \mu) induces a process R_n o on the space X, where o is a base point in X. We are interested in the deviations of the normalized distance to the origin d(R_n o, o)/n around the drift (its almost sure limit) in two different directions:
1-Finite time estimations: Hoeffding type subgaussian concentration bounds of the probability that d(R_n o, o)/n deviates around the drift. The constants implied in our inequality depend, as in the scalar case, on the size of the support; but also on the norm of $\mu$ in the regular representation of G. Particular attention will be given to random walks on Fuchsian groups. Application to different versions of the Tits alternative on these spaces is given.
2-Asymptotic result: local behavior of the limit Log Laplace; and link with the CLT established by Benoist-Quint in this setting. Dually this gives local information on the rate function of large deviations shown in this context by Boulanger--Mathieu--Sert--Sisto.
Joint works with C. Sert and P. Mathieu et C. Sert.
Tuesday January 10
L'ENS, salle W, room W
The seminar will also be streamed via Zoom
14.00 -- 14.45 Greg Kuperberg (UC Davis & IHES) "An obstruction to quantum algorithms from small cancellation theory"
15.00 -- 15.45 Konstantinos Tsouvalas (IHES ) "Linear hyperbolic groups indiscrete in rank 1 and products "
16.00-16.45 Delaram Kahrobaei (Cuny & IHES) "Applied Group Theory in the Quantum and Artificial Intelligence Era"
Greg Kuberberg, "An obstruction to quantum algorithms from small cancellation theory".
Shor's algorithm, which is one of the most important results in theoretical quantum computing, miraculously finds the period of a periodic function on the integers, in quantum polynomial time in the number of digits of the answer. The problem that Shor's algorithm solves generalizes to the hidden subgroup problem, whereby a function f on a group G is H-periodic for some unknown subgroup H, and the problem is to calculate H given access to f. This problem varies tremendously depending on both G and H; it is seen as harder when G is highly noncommutative, but easier when H is normal. I will discuss my result that if G is a non-abelian free group and H is assumed to be normal, then the hidden subgroup problem is NP-hard and a Shor-type algorithm is implausible at best. The proof depends on the structure theory of small-cancellation groups. Among other things, I will establish and use a fast algorithm to compute a merged form for all geodesic words for an arbitrary group element in a small-cancellation group.
Konstantinos Tsouvalas "Linear hyperbolic groups indiscrete in rank 1 and products ".
Gromov hyperbolic groups is a rich class of finitely presented groups introduced by Gromov in the 80s, capturing the coarse geometric properties of fundamental groups of closed negatively curved Riemannian manifolds. While there are certain classes of hyperbolic groups which can be realized as discrete subgroups of some general linear group over C (e.g. Anosov groups), there are no known linear examples which fail to admit discrete faithful complex representations. In this talk, we are going to provide constructions of linear hyperbolic groups which fail to admit discrete faithful representation into any semisimple Lie group of rank 1 or in products of rank 1 Lie groups. These are joint works with Sami Douba and Nicolas Tholozan.
Delaram Kahrobaei "Applied Group Theory in the Quantum and Artificial Intelligence Era"
In this talk I present an overview of the current state-of-the-art in post-quantum group-based cryptography. I describe several families of groups that have been proposed as platforms,with special emphasis in polycyclic groups and graph groups, dealing in particular with their algorithmic properties and cryptographic applications. I then describe some applications of combinatorial algebra in fully homomorphic encryption, and in particularhomomorphic machine learning. In the end I will discuss several open problems in this direction.
Tuesday December 20
L'ENS, salle W, room W
The seminar will also be streamed via Zoom:
https://us02web.zoom.us/j/8921997963
Password: Consider a Cayley graph of a free group on 107 generators (with respect to this free generating set). What is the degree of this graph? Type the three digit number as the password.
14.00 -- 14.45 Jim Belk (University of Glasgow), "Embeddings into Finitely Presented Simple Groups" (in presence, room W)
15.00 -- 15.45 Mark Pengitore (University of Virginia), "Characteristic quotients of surface groups and residual finiteness of mapping class
groups" (online, shown on the screen in Room W)
16.00 -- 16.45 Nikolay Bogachev (Weizmann Institute, IITP RAS), "On geometry
and arithmetic of hyperbolic orbifolds" (TBC, online, shown on the screen room W)
Tuesday, November 22
L'ENS, salle W
The seminar will also be streamed via Zoom.
https://us02web.zoom.us/j/86001581637
Meeting ID: 860 0158 1637
Password: Consider a Cayley graph of a free group on 107 generators (with
respect to this free generating set). What is the degree of this graph?
Type the three digit number as the password.
14.00-14.45 Anne Lonjou (UPV/EHU, University of the Basque Country).
15.00 -15.45 Eduardo Silva (ENS, Paris)
16.00 - 16.45 Alina Vdovina (CUNY, New York)
Anne Lonjou "Cremona goup over finite fields and Neretin groups". The Cremona group, the group of birational transformations of the projective plane, is a group coming from algebraic geometry. The Neretin group, the group of almost automorphisms of a regular rooted tree, is coming from topology of low dimension. Even if these two families of groups are different, we will explain in this talk how they are linked and the properties that we can obtain with this point of view. This is a joint work with Anthony Genevois and Christian Urech.
Eduardo "Silva Dead ends on wreath products and lamplighter groups".
A finitely generated group G has unbounded depth with respect to a finite generating set S if for any n≥1, there exists g in G such that multiplying g by any word of S-length at most n results in an element of word length at most |g|_S. In other words, g locally maximizes the word length | |_S in its n-neighborhood.In this talk we will concentrate on the case where G=A wr B is the wreath product of two groups. We prove that for any finite group A and any finitely generated group B, the group A wr B admits a standard generating set with unbounded depth, and that if B is abelian then the above is true for every standard generating set. This generalizes the case where B = Z together with its cyclic generator, due to Cleary and Taback. When B = H * K is the free product of two finite groups H and K, we characterize which standard generators of the associated wreath product have unbounded depth in terms of a geometrical constant related to the Cayley graphs of H and K. In particular, our result shows a difference with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups does not have unbounded depth with respect to some standard generating set.
Alina Vdovina "Higher structures in mathematics: buildings, k-graphs and C*-algebras".
We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k.
The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.The talk is based on joint papers with Nadia Larsen, Sam Mutter, Christiana Radu.
Wendesday, October 19 (unusual day)
L'ENS, salle W
The seminar will also be streamed via Zoom.
14.00-14.45 Yuri Neretin (University of Vienna)
15.00 -15.45 Matteo Tarocchi (University of Milano-Bicocca )
16.00 - 16.45 Rachel Skipper (ENS, Paris)
Yuri Neretin, "Infinite symmetric groups and cobordisms of triangulated surfaces". Let $G$ be the product of three copies of the infinite symmetric group $S(\infty)$. The diagonal subgroup $K$ of $G$ and stabilizers $K(n) ⊂ S(\infty)$ of the first $n$ points. We show that double coset spaces $K(m) \ G/K(n)$ form a category and this category acts in a natural way in unitary representations of $G$. This category admits a description in terms of concatenation cobordisms of colored triangulated surfaces and the set $K(0) \ G/K(0)$ is in one-to-one correspondence with Belyi data.
Matteo Tarocchi, "Thompson-like groups acting on fractals". Introduced in the '60s by Richard Thompson, each of the three Thompson groups F, T and V has made its appearance in many different topics. The groups T and V were the first examples of infinite finitely presented simple groups, whereas the fame of its smaller sibling F mostly originates from the decades-old open question regarding its possible amenability. In 2019 J. Belk and B. Forrest introduced a generalization of Thompson groups, the family of Rearrangement Groups. These are groups of certain homeomorphisms of fractals that act by permuting the self-similar pieces that make up the fractal. This talk will introduce Thompson groups and Rearrangement Groups, highlighting some known facts about them, such as the simplicity of the commutator subgroups of the Basilica and Airplane rearrangement groups and a general result about invariable generation.
Rachel Skipper, "Maximal Subgroups of Thompson's group V". Maximal subgroups of a group provide a range of information about the group. First, maximal subgroups correspond to primitive actions of a group. Secondly, in a finitely generated group every proper subgroup is contained in a maximal one. In this talk, we will discuss some ongoing work with Jim Belk, Collin Bleak, and Martyn Quick to understand and classify maximal subgroups of Thompson's group V.
2021-2022
Organized by Anna Erschler, Nima Hoda et Ivan Mitrofanov
Supported by the ERC grant GroIsRan
videos of the previous talks
https://www.math.ens.psl.eu/~frisch/seminar_video/
Tuesday, March 15
L'ENS, salle W
The seminar will also be streamed via Zoom:
https://us02web.zoom.us/j/82070470538
Meeting ID: 820 7047 0538
Password: Consider a Cayley graph of a free group on 107 generators (with respect to this free generating set). What is the degree of this graph? Type the three digit number as the password.
14.00-14.45 Marcin Sabok (McGill University)
15.00 -15.45 Juan Paucar (Jussieu)
16.00 - 16.45 Josh Frisch (ENS, Paris)
Marcin Sabok, "Hyperfiniteness at hyperbolic boundaries". I will discuss recent results establishing hyperfiniteness of the equivalence relations induced by actions on the Gromov boundaries of various hyperbolic spaces. This includes boundary actions of hyperbolic groups (joint work with T. Marquis) and actions of the mapping class group on the boundaries of the arc graph and the curve graph (joint work with P. Przytycki).
Juan Paucar, "Coarse embeddings between locally compact groups and quantitative measured equivalence". I will discuss about quantitative versions of Measure Equivalence for locally compact compactly generated groups, a notion introduced by Tessera, Le Maître, Delabie and Koivisto on the finitely generated case. Moreover, they introduced as well quantitative asymmetric versions of it, called L^p-measured subgroups, and in particular they proved that coarse embeddings between amenable groups imply the existence of a L∞-measured coupling. In this talk, I will prove the same statement on the locally compact case, which will gives us an obstruction to coarse embeddings for locally compact compactly generated groups.
Josh Frisch, "Characteristic Measures and Minimal Subdynamics". Given a topological dynamical system (a group G acting by homeomorphisms on a compact space X) a measures on X is said to be characteristic if it is invariant to the automorphism group of the system. A system is called minimal if it has no closed G invariant subsystems. In this talk I will give a brief introduction to characteristic measures before explaining the main result: a minimal dynamical system without characteristic measures. This is joint work with Brandon Seward and Andy Zucker
Tuesday, February 8,
on ZOOM
ZOOM: https://us02web.zoom.us/j/81548053762
ID: 815 4805 3762
Password: Consider a Cayley graph of a free group on 107 generators
(with respect to this free generating set). What is the degree of
this graph? Type the three digit number as the password.
An afternoon on invariant and stationary random subgroups.
15.00-15.45 Tsachik Gelander (Weizmann Institute)
16.00 -16.45 Matthieu Joseph (ENS Lyon)
17.15 - 18.00 Yair Hartman (Ben Gurion University)
Tsachik Gelander, "Stationary random discrete subgroups of semisimple Lie groups ". The theory of IRS (invariant random subgroups) has proven to be very useful to the study of lattices and their asymptotic invariants. However, restricting to invariant measures (on the space of subgroups) is a big compromise (since our groups are non-amenable) and limits the scope of problems that one can investigate. I will explain a fundamental inequality concerning the discreteness function, proved in a joint work with G.A. Margulis and A. Levit, which allows extending various results about IRS to SRS (stationary random subgroups). The same inequality gives control on certain random walks on the space ofdiscrete subgroups. Finally I will outline the proof obtained jointly with M. Fraczyk of the following conjecture of Margulis: Let G be a higher rank simple Lie group and D a discrete subgroup. If D is confined, i.e. if there is a compact set in G\{1} which meets every conjugate of D, then D is a lattice in G. This result gives a far reaching generalisation of the celebrated normal subgroup theorem of Margulis.
Matthieu Joseph, "Allosteric actions of surface groups". In a recent work, I introduced the notion of allosteric actions: a minimal action of a countable group on a compact space, with an ergodic invariant measure, is allosteric if it is topologically free but not essentially free. In the first part of my talk I will explain some properties of allosteric actions, and their links with Invariant Random Subgroups (IRS). In the second part, I will explain a recent result of mine: the fundamental group of a closed hyperbolic surface admits allosteric actions.
Yair Hartman, "Intersectional Invariant Random Subgroups"
Tuesday, January 11,
on ZOOM
15.00 - 15.45 Friedrich Martin Schneider (Freiberg)
16.00 -16.45 Eduardo Scarparo (Federal University of Santa Catarina)
17.15 - 18.00 Gidi Amir (Bar Ilan)
Friedrich Martin Schneider, "Concentration of invariant means"
In the context of large (non-locally compact) topological groups, one frequently witnesses an extreme form of amenability: extreme amenability. A topological group G is called extremely amenable if everycontinuous action of G on a non-void compact Hausdorff space admits afixed point. Most of the currently known manifestations of this phenomenon have been exhibited using either structural Ramsey theory, or concentration of measure. The talk will be focused on the latter method. Among other things, I will discuss a new concentration result for convolution products of invariant means, based on a suitable adaptation of Azuma's inequality. Furthermore, I will show how this result can beused to prove extreme amenability of certain topological groups arising from von Neumann's continuous geometries.
Eduardo Scarparo "Amenability and unitary representations of groups of dynamical origin.
In the first half, we report on joint work with Mehrdad Kalantar in which we completely characterize C*-simplicity of quasi-regular representations associated to stabilizers of boundary actions in terms of amenability of the isotropy groups of the groupoid of germs of the action. For quasi-regular representations associated to "open" stabilizers, a complete characterization of C*-simplicity is still missing, and we illustrate this fact with an ad hoc proof that, for Thompson's group F < T, the quasi-regular representation of T associated to [F,F] properly weakly contains the one associated to F (a year ago Kalantar spoke at this seminar and I will emphasize the new results and examples obtained since then).In the second half, we show that the topological full group of a minimal action on the Cantor set is C*-simple if and only if the alternating full group is non-amenable. We use this to conclude that, e.g., for free actions of groups of subexponential growth, non-amenability of the topological full group is equivalent to C*-simplicity, but in general this equivalence is an open problem.
Gidi Amir "Amenability of quadratic activity automata groups".
Automata groups are a family of groups acting on rooted trees that have a simple definition yet exhibit a very rich behavior. Automaton groups include many interesting examples such as Grigorchuk groups, the Basilica group, Hanoi tower groups and lamplighter groups.
The activity of an automaton group, introduced by Sidki, can be viewed as a measure of complexity that can grow either polynomially (with some degree) or exponentially. Sidki proved that polynomial activity automata groups do not contain free subgroups, which prompted him to ask “Are all polynomial activity automata groups amenable?”
This was answered positively for degree 0 (“bounded”) by Bartholdi-Kaimanovich-Nekrashevych and for degree 1 (“linear”) by Amir-Angel-Virag.
Juschenko, Nekrashevych and de la Salle gave a general approach allowing to deduce the amenability of groups from recurrence of the orbital Schreier graphs of group actions satisfying some conditions. This allowed, among other things, to reprove the amenability of automata groups of degree 0 and 1, and to prove the conditional result that if the "natural" action of a quadratic activity (d=2) automata group is recurrent then it is amenable.
In recent work with Omer Angel and Balint Virag, we prove that the natural Schreier graphs of the quadratic activity mother groups, a special family into which all quadratic activity automata groups can be embedded, is recurrent. This allows us to conclude the amenability of all quadratic activity automata groups.The proof relies on bounding the electrical resistance between vertices in the Schreier graphs, which in turn relies on a "combinatorial" analysis of the graph structure together with new Nash-Williams type lower bound on resistances.
After surveying some background on automata groups, mother groups and electrical resistance, and some previous amenability results on automata groups, we will focus on the new analysis giving the resistance lower bounds. No previous knowledge on random walks, automata groups or electrical resistance will be assumed. This talk is based on joint work with O. Angel and B. Virag.
Tuesday, December 14
on ZOOM
15.00 - 15.45 MurphyKate Montee (Carleton College)
16.00 -16.45 Tsung-Hsuan Tsai (IRMA, Strasbourg)
17.15 - 18.00 Damian Orlef (IMPAN, Warsaw)
MurphyKate Montee, "Cubulating Rand om Groups at Densities d<3/14"
Random groups are one way to study "typical" behavior of groups. In the Gromov density model, we often find a threshold density above which a property is satisfied with probability 1, and below which it is satisfied with probability 0. Two properties of random groups that have studied are cubulation and Property (T). In this setting these are mutually exclusive, but the threshold densities are not known. In this talk I'll present a method to demonstrate cubulation on groups with density less than 3/14, and discuss how this might be extended to demonstrate cubulation for densities up to 1/4. In particular, I will describe a construction of walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex.
This extends results of Ollivier-Wise and Mackay-Przytycki at densities less than 1/5 and 5/24, respectively.
Tsung-Hsuan Tsai, "Freiheitssatz for the density model of random groups"
Magnus' Freiheitssatz states that if a group is defined by a presentation with m generators and a single cyclically reduced relator, and this relator contains the last generating letter, then the first m-1 letters freely generate a free subgroup. We study an analogue of this theorem in the Gromov density model of random groups, showing a phase transition phenomenon at density d_r = min{1/2, 1-log_{2m-1}(2r-1)} with 0<r<m: we prove that for a random group with m generators at density d, if d<d_r then the first r letters freely generate a free subgroup; whereas if d>d_r then the first r letters generate the whole group.
Damian Orlef (IMPAN, Warsaw), "Non-orderability of random triangular groups by using random 3CNF formulas"
A random group in the triangular binomial model Gamma(n,p) is given by the presentation < S|R >, where S is a set of n generators and R is a random set of cyclically reduced relators of length 3 over S, with each relator included in R independently with probability p. When n tends to infinity, the asymptotic properties of groups in Gamma(n,p) vary widely with the choice of p=p(n). By Antoniuk-Łuczak-Świątkowski and Żuk, there existconstants C, C', such that a random triangular group is asymptotically almost surely (a.a.s.) free if p<Cn^{-2} and a.a.s.infinite, hyperbolic, but not free, if p in (C'n^{-2}, n^{-3/2-\varepsilon}). We generalize the second statement by findinga constant c such that if p\in(cn^{-2}, n^{-3/2-\varepsilon}), then a random triangular group is a.a.s. not left-orderable. We prove this by linking left-orderability of Gamma in Gamma(n,p) to thesatisfiability of the random propositional formula, constructed from the presentation of Gamma. The left-orderability of quotients will be also discussed.
Monday, November 15 (Attention: unusual day!)
l'ENS, salle W
14.00 - 14.45 Piotr Przytycki
15.00- 15.45 Sami Douba
16.00- 16.45 Jean Lecureux
Piotr Przytycki, "Groups acting almost freely on 2-dimensional CAT(0) complexes satisfy the Tits Alternative"
Let X be a 2-dimensional complex with piecewise smooth Riemannian metric, finitely many isometry types of cells, that is CAT(0). Let G be a group acting on X with a bound on cell stabilisers. We will sketch the proof of the Tits Alternative saying that G is virtually cyclic, virtually Z^2 or contains a nonabelian free group. This generalises our earlier work for X a 2-dimensional systolic complex or a 2-dimensional Euclidean building. This is joint work with Damian Osajda.
Sami Douba "Proper CAT(0) actions of unipotent-free linear groups".
Button observed that finitely generated matrix groups containing no nontrivial unipotent matrices behave much like groups admitting proper actions by semisimple isometries on complete CAT(0) spaces. It turns out that any finitely generated matrix group possesses an action on such a space whose restrictions to unipotent-free subgroups are in some sense tame. We discuss this phenomenon and some of its implications for the representation theory of certain 3-manifold groups.
Jean Lecureux, "Rigidity properties of Ã_2 lattices".
Buildings of type Ã_2 are commonly associated to groups such as G=SL_3(k), where k is a non-archimedean local field. Lattices in such a group G have strong rigidity properties (for example, they satisfy Margulis' superrgidity). But there are also buildings for which the automorphism group is smaller, and much less understood - but in some cases still cocompact. In this talk I will explain how these other "exotic" lattices are still very rigid, and raise some open questions.
Monday, October 11 (Attention: unusual day!)
l'ENS, salle W
14.00 - 14.45 François Le Maître (Université Paris Diderot -Paris VII)
15.00- 15.45 Romain Tessera (Université Paris Diderot -Paris VII)
16.00- 16.45 Pierre Fima (Université Paris Diderot -Paris VII)
François Le Maître "Reconstruction for Boolean measure-preserving actions of full groups and applications"
Given a two measure-preserving ergodic action of countable groups on a standard probability space, Dye's reconstruction theorem asserts that any isomorphism between the associated full groups must come from an isomorphism of the space which sends the first partition of the space into orbits to the second. It is thus natural to ask what happens more generally for homomorphisms between full groups. I will present a joint work with Alessandro Carderi and Alice Giraud where we show that any such homomorphism comes from a measure-preserving action of the equivalence relation or of one of its symmetric powers. Such a result is very similar in spirit to Matte Bon's striking classification of actions by homeomorphisms of topological full groups, but we will see that the proof is much simpler modulo the Thomas-Tucker-Drob classification of invariant random subgroups of the dyadic symmetric group. As an application, we characterize Kazhdan's property (T) of a measure-preserving equivalence relation in terms of its full group: the equivalence relation has (T) if and only if all non-free ergodic Boolean actions of its full group are strongly ergodic.
Romain Tessera "Coarse geometry meets measured group theory" .
We will present a new induction technique based on ideas of Gromov and Shalom. Given two finitely generated groups H and G and a Lipschitz injective map from H to G, we construct a topological coupling space between them. If H is amenable, then this enables us to view H as a ``measured subgroup" of G. Using this formalism, we manage to prove that the Folner function of G grows faster than the Folner function of H.
An application of this result is the following (new) theorem: an amenable group coarsely embeds into a hyperbolic group if and only it is virtually nilpotent.
Pierre Fima, "Highly transitive groups among groups acting on trees".
After an introduction to the topic of highly transitive groups, I will present a joint work with F. Le Maître, S. Moon and Y. Stalder in which we characterize groups acting on trees which are highly transitive.
Tuesday, June 22
An afternoon on quasi-isometries of groups.
https://us02web.zoom.us/j/83038016716
Password: Consider a Cayley graph of a free group on 107 generators
(with respect to this free generating set). What is the degree of
this graph? Type the three digit number as the password.
15.30 - 16.15 Chris Hruska (University of Wisconsin)
16.30 - 17.15 Anthony Genevois (Montpellier)
17.45 - 18.30 Romain Tessera (Jussieu)
Chris Hruska, "Canonical splittings of relatively hyperbolic groups"
A JSJ decomposition is a graph of groups decomposition that allows one to classify all splittings of a group over certain subgroups. I will discuss a JSJ decomposition for relatively hyperbolic groupssplitting over elementary subgroups that depends only on the topology of its boundary. This decomposition could potentially be of use forunderstanding groups that have homeomorphic boundaries, but are not necessarily quasi-isometric. (Joint work with Matt Haulmark.)
Anthony Genevois "Asymptotic geometry of lamplighters over one-ended groups".
After a general introduction to lamplighter groups and their asymptotic geometry, I will describe a complete quasi-isometric classification of lamplighters over one-ended finitely presented groups. The proof will be briefly overviewed, and the rest of the talk will be dedicated to the central tool of the argument: an embedding theorem proved thanks to (quasi-)median geometry.
Romain Tessera "Asymptotic geometry of lamplighters over one-ended groups II".
This second talk will be dedicated to the asymmetry between amenable and non-amenable groups in the quasi-isometric classification previously described. In particular, I will explain why lamplighters over non-amenable groups are more often quasi-isometric than lamplighters over amenable groups. Also, I will show how the distance from a quasi-isometry between amenable groups to a bijection can be quantified, introducing quasi-k-to-one quasi-isometries for an arbitrary real k>0, and explain how this notion is fundamental in the understanding of the asymptotic geometry of lamplighters over amenable groups.
Tuesday, May 25
ZOOM: https://us02web.zoom.us/j/85750528881
Meeting ID: 857 5052 8881
Password: Consider a Cayley graph of a free group on 107 generators
(with respect to this free generating set). What is the degree of this
graph? Type the three digit number as the password.
15.30 - 16.15 Giulio Tiozzo (Toronto)
16.30 - 17.15 Sébastien Gouëzel (Rennes)
17.45 - 18.30 Andrei Alpeev (St-Petersburg)
Giulio Tiozzo, "The fundamental inequality for cocompact Fuchsian groups".
A recurring question in the theory of random walks on hyperbolic spaces asks whether the hitting (harmonic) measures can coincide with measures of geometric origin, such as the Lebesgue measure. This is also related to the inequality between entropy and drift.
For finitely-supported random walks on cocompact Fuchsian groups with symmetric fundamental domain, we prove that the hitting measure is singular with respect to Lebesgue measure; moreover, its Hausdorff dimension is strictly less than 1.
Along the way, we prove a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.
Joint with P. Kosenko.
Sébastien Gouëzel, "Exponential estimates for random walks without moment conditions on
hyperbolic spaces"
Consider a random walk on a nonelementary hyperbolic space (proper or not, but one may just think of a free group for simplicity). It is known
that the walk is converging almost surely towards a point at a boundary, and that the rate of escape is positive. We will discuss quantitative
versions of these statements: when can one show that these facts hold with an exponentially small probability for exceptions? While there are
several such results in the literature, the originality of our approach is that it does not require any moment condition on the random walk. We
will discuss the main technical new idea in the case of the free group.
Andrei Alpeev, "Examples of different boundary behaviour of left and right random walks on groups".
In 80-s Vadim Kaimanovich presented a construction of a non-degenerate measure on the standard lamplighter group which has trivial right random walk boundary and non-trivial left random walk boundary. I will show that examples of such kind are possible exactly for amenable groups with non-trivial ICC factors.
Tuesday, April 27
15.00 - 15.45 Panos Papazoglu (Oxford)
16.00 - 16.45 Urs Lang (ETH Zurich)
17.15 - 18.00 Karim Adiprasito (Hebrew University & University of Copenhagen)
Panos Papazoglu, "Asymptotic dimension of planes" (joint with K. Fujiwara)
It is easy to see that there are Riemannian manifolds homeomorphic to $\mathbb R ^3$
with infinite asymptotic dimension. In contrast to this we showed with K. Fujiwara that
the asymptotic dimension of Riemannian planes (and planar graphs) is bounded by 3. This was
improved to 2 by Jorgensen-Lang and Bonamy-Bousquet-Esperet-Groenland-Pirot-Scott.
Urs Lang, "Assouad-Nagata dimension and Lipschitz extensions "
It follows from a recent result of Fujiwara-Papasoglu and a Hurewicz-type theorem due to Brodskiy-Dydak-Levin-Mitra that every planar geodesic metric space has
(Assouad-)Nagata dimension at most two and hence asymptotic dimension at most two. This can be used further to prove that every three-dimensional Hadamard manifold
has Nagata dimension three and is an absolute Lipschitz retract (joint work with Martina Jørgensen). The role of the Nagata dimension in Lipschitz extension problems
will be discussed further.
Karim Adiprasito, "l^2 cohomology and stable Lefschetz theory"
Tuesday March 30
https://us02web.zoom.us/j/85927181837
Password: Consider a Cayley graph of a free group on 107 generators (with respect to this free generating set). What is the degree of this graph? Type the three digit number as the password.
14.00 - 14.45 Hanna Oppelmayer (TU Graz)
15.00 - 15.45 Georgii Veprev (St-Petersburg)
16.15 - 17.00 Paul-Henry Leemann (University of Neuchâtel)
Hanna Oppelmayer, "Random walks on dense subgroups of totally disconnected locally compact groups"
There is a class of random walks on some countable discrete groups that capture the asymptotic behaviour of certain random walks
on totally disconnected locally compact second countable (t.d.l.c.) groups which are completions of the discrete group. We will see that
the Poisson boundary of the t.d.l.c. group is always a factor of the Poisson boundary of the discrete group, when equipped with these
random walks. All this is done by means of a so-called Hecke subgroup.
In particular, if the two Poisson boundaries are isomorphic then this Hecke subgroup is forced to be amenable. The reverse direction holds
whenever there is a uniquely stationary compact model for the Poisson boundary of the discrete group. Furthermore, we will deduce some
applications to concrete examples, like the lamplighter group over Z and solvable Baumslag-Solitar groups and show that they are prime,
i.e. there are random walks such that the Poisson boundary and the one-point-space are the only boundaries.
This is a joint work with Michael Björklund (Chalmers, Sweden) and
Yair Hartman (Ben Gurion University, Israel).
Georgi Veprev, "Non-existence of a universal zero entropy system for non-periodic amenable group actions"
Let G be a discrete amenable group. We study interrelations between topological and measure-theoretic actions of G. For a given continuous representation of G on a compact metric space X we consider the set of all ergodic invariant measures on X. For any such measure we associate the corresponding measure-theoretic dynamical system. The general wild question is what the family M of these systems could be up to measure-theoretic isomorphisms.
The topological system for which M coincides with a given class S of ergodic actions is called universal. B.Weiss's question regards the existence of a universal system for the class of all zero-entropy actions. For the case of Z, the negative answer was given by J. Serafin.
Our main result establishes the non-existence of a universal zero-entropy system for any non-periodic amenable group. The condition of non-periodicity is crucial in our arguments so the question is still open for general torsion amenable groups.
Our proof bases on the slow entropy type invariant called scaling entropy introduced by A. Vershik. This invariant characterizes the intermediate growth of the entropy in a sense on the verge of topological and measure-preserving dynamics. I will present a brief survey of scaling entropy and show how this invariant applies to the non-existence theorem.
Paul-Henry Leemann, "De Bruijn graphs, spider web graphs and Lamplighter groups"
De Bruijn graphs represent word overlaps in symbolic dynamical systems. They naturally occur in dynamical systems and combinatorics, as well as in computer science and bioinformatics. We will show that de Bruijn graphs converge to a Cayley graph of the Lamplighter group and and will also compute their spetra. We will then discuss some generalizations of them as for examples Spider web graphs or Rauzy graphs.
Based on a joint work with R. Grigorchuk and T. Nagnibeda.
Tuesday February 23
15.30 - 16.15 Jingyin Huang (Ohio State University)
16.30 -17.15 Jérémie Chalopin (Aix-Marseille Université)
17.45 - 18.30 Daniel Wise (McGill University)
Jingyin Huang "Morse quasiflats".
We are motivated by looking for traces of hyperbolicity in a space or group which is not Gromov-hyperbolic. One previous approach in this
direction is the notion of Morse quasigeodesics, which describes ``negatively-curved'' directions in the spaces; another previous
approach is ``higher rank hyperbolicity'' with one example being that though triangles in products of two hyperbolic planes are not thin,
tetrahedrons made of minimal surfaces are ``thin''. We introduce the notion of Morse quasiflats, which unifies these two seemingly
different approaches and applies to a wider range of objects. In the talk, we will provide motivations and examples for Morse quasiflats,
as well as a number of equivalent definitions and quasi-isometric invariance (under mild assumptions). We will also show that Morse
quasiflats are asymptotically conical, and comment on potential applications. Based on joint work with B. Kleiner and S. Stadler.
Jérémie Chalopin , "Event structures, median graphs and CAT(0) cube complexes".
Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these
structures as combinatorial and geometric objects and both conjectures can be reformulated in this framework. Namely, from a graph theoretical
point of view, the domains of prime event structures correspond exactly to median graphs; from a geometric point of view, these domains are in
bijection with CAT(0) cube complexes.
Thiagarajan conjectured that regular event structures correspond exactly to event structures obtained as unfoldings of finite 1-safe Petri nets.
Using the bijections between event structures, median graphs and CAT(0) cube complexes, we disproved this conjecture.
Our counterexample is derived from an example by Wise of a nonpositively curved square complex whose universal cover is a CAT(0) square complex
containing a particular plane with an aperiodic tiling.
On the positive side, we show that event structures obtained as unfoldings of finite 1-safe Petri nets correspond to the finite special
cube complexes introduced by Haglund and Wise.
Daniel Wise, "Complete Square Complexes".
A "Complete Square Complex” is a 2-complex X whose universal cover is the product of two trees.
Obvious examples are when X is itself the product of two graphs but there are many other examples.
I will give a quick survey of complete square complexes with an aim towards describing some problems about them
and describing some small examples that are “irreducible” in the sense that they do not have a finite cover that is a product.
Tuesday January 19
ZOOM: https://us02web.zoom.us/j/84778703586
ID: 847 7870 3586
Password: Consider a Cayley graph of a free group on 107 generators
(with respect to this free generating set). What is the degree of
this graph? Type the three digit number as the password.
16.00-16.45 Igor Pak (UCLA)
17.00-17.45 Behrang Forghani (the College of Charleston)
18.15-19.00 Mehrdad Kalantar (University of Houston)
Igor Pak "Cogrowth sequences in groups and graphs"
Let G be a finitely generated group with generating set S. We study the cogrowth sequence {a_n(G,S)}, which counts the number of words of length n over the alphabet S that are equal to 1 in G. I will survey rеcent asymptotic and analytic results on the cogrowth sequence, motivated by both combinatorial and algebraic applications. I will then present our recent work with Kassabov on spectral radii of Cayley graphs, which are also governed by the asymptotics of cogrowth sequences.
Behrang Forghani "Boundary Preserving Transformations"
This talk concerns the situations when the Poisson boundaries of different random walks on the same group coincide. In some special cases, Furstenberg and Willis addressed this question. However, the scopes of their constructions are limited. I will show how randomized stopping times can construct measures that preserve Poisson boundaries and discuss their applications regarding the Poisson boundary identification problem. This talk is based on joint work with Kaimanovich.
Mehrdad Kalantar "On weak containment properties of quasi-regular representations of stabilizer subgroups of boundary actions"
A continuous action of a group G on a compact space X is said to be a boundary action if the weak*-closure of the orbit of every Borel probability on X under G-action contains all point measures on X. Given a boundary action of a discrete countable group, we prove that at any continuity point of the stabilizer map, the quasi-regular representation of the stabilizer subgroup is weakly equivalent to every representation that it weakly contains. We also completely characterize when these quasi-regular representations weakly contain the GNS representation of a character on the group.
This is joint work with Eduardo Scarparo.
Tuesday December 8,
in ZOOM
15.00-15.45 Robert Young (NYU Courant and IAS Princeton)
16.00-16.45 Matei Coiculescu (Brown University)
17.15-18.00 Richard Schwartz (Brown University and IAS Princeton)
Robert Young, "Hölder maps to the Heisenberg group"
In this talk, we construct Hölder maps to the Heisenberg group H, answering a question of Gromov. Pansu and Gromov observed that any surface embedded in H has Hausdorff dimension at least 3, so there is no α-Hölder embedding of a surface into H when α > 2/3. Züst improved this result to show that when α > 2/3, any α-Hölder map from a simply-connected Riemannian manifold to H factors through a metric tree. We use new techniques for constructing self-similar extensions to show that any continuous map to H can be approximated by a (2/3 - ε)-Hölder map. This is joint work with Stefan Wenger.
Matei Coiculescu, "The Spheres of Sol".
Sol, one of the eight Thurston geometries, is a solvable three-dimensional Lie group equipped with a canonical left invariant metric. Sol has sectional curvature of both signs and is not rotationally symmetric, which complicates the study of its Riemannian geometry.
Our main result is a characterization of the cut locus of Sol, which implies as a corollary that the metric spheres in Sol are topological spheres.
This is joint work with Richard Schwartz".
Richard Schwartz, "The areas of metric spheres in Sol".
This is a sequel talk, following Matei Coiculescu's talk about our joint work characterizing the cut locus of the identity in Sol.
In this talk, I will explain my result that the area of a metric sphere of radius r in Sol is at most Ce^r for some uniform constant C. That is,
up to constants, the sphere of radius r in Sol has the same area as the hyperbolic disk of radius r.
Tuesday, November 24, in ZOOM
14.00-14.45 Alessandro Sisto (Heriot-Watt)
15.00-15.45 Thomas Haettel ( Montpellier)
16.15-17.00 Mark Hagen (Bristol)
Alessandro Sisto "Cubulation of hulls and bicombings"
It is well-known that the quasi-convex hull of finitely many points in a
hyperbolic space is quasi-isometric to a tree. I will discuss an
analogous fact in the context of hierarchically hyperbolic spaces, a
large class of spaces and groups including mapping class groups,
Teichmueller space, right-angled Artin and Coxeter groups, and many
others. In this context, the approximating tree is replaced by a CAT(0)
cube complex. I will also briefly discuss applications, including how
this can be used to construct bicombings.
Based on joint works with Behrstock-Hagen and Durham-Minsky.
Thomas Haettel "The coarse Helly property, hierarchical hyperbolicity and semihyperbolicity"
For any hierarchical hyperbolic group, and in particular any mapping
class group, we define a new metric that satisfies a coarse Helly
property. This enables us to deduce that the group is semihyperbolic,
i.e. that it admits a bounded quasigeodesic bicombing, and also that
it has finitely many conjugacy classes of finite subgroups. This has
several other consequences for the group. This is a joint work with
Nima Hoda and Harry Petyt.
Mark Hagen "Wallspaces, the Behrstock inequality, and l_1 metrics on
asymptotic cones"
From its hyperplanes, one can always characterise a CAT(0)
cube complex as the subset of some (often infinite) cube consisting of
the solutions to a system of "consistency" conditions. Analogously, a
hierarchically hyperbolic space (HHS) can be coarsely characterised as a
subset of a product of Gromov-hyperbolic spaces consisting of the
"solutions" to a system of coarse consistency conditions.
HHSes are a common generalisation of hyperbolic spaces, mapping class
groups, Teichmuller space, and right-angled Artin/Coxeter groups. The
original motivation for defining HHSes was to provide a unified
framework for studying the large-scale properties of examples like these.
So, it is natural to ask about the structure of asymptotic cones of
hierarchically hyperbolic spaces.
Motivated by the above characterisation of a CAT(0) cube complex, we
introduce the notion of an R-cubing. This is a space that can be
obtained from a product of R-trees, with the l_1 metric, as a solution
set of a similar set of consistency conditions. R-cubings are therefore
a common generalisation of R-trees and (finite-dimensional) CAT(0) cube
complexes. R-cubings are median spaces with extra structure, in much
the same way that HHSes are coarse median spaces with extra structure.
The main result in this talk says that every asymptotic cone of a
hierarchically hyperbolic space is bilipschitz equivalent to an
R-cubing. This strengthens a theorem of Behrstock-Drutu-Sapir about
asymptotic cones of mapping class groups. Time permitting, I will talk
about an application of this result which is still in progress, namely
uniqueness of asymptotic cones of various hierarchically hyperbolic
groups, including mapping class groups and right-angled Artin groups.
This is joint work with Montse Casals-Ruiz and Ilya Kazachkov.
Tuesday, October 27, in ZOOM, attention: unusual time!
09.00 - 09.45 Koji Fujiwara (Kyoto)
10.00 - 10.45 Macarena Arenas (Cambridge)
11.15-12.00 Indira Chatterji (Nice)
Koji Fujiwara "The rates of growth in a hyperbolic group"
Macarena Arenas "Linear isoperimetric functions for surfaces in hyperbolic groups"
One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for
a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for
disc diagrams D -->X. It is likewise known that hyperbolic groups have a linear annular
isoperimetric function and a linear homological isoperimetric function. I will tell you a bit about these isoperimetric functions
and a generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.
Indira Chatterji "Tangent bundles on hyperbolic spaces and proper actions on Lp spaces".
I will define a notion of a negatively curved tangent bundle of a metric measured space, and relate that notion to proper actions on Lp spaces. I will discuss hyperbolic spaces as examples.
2019-2020
Tuesday, June 30, in ZOOM,
14.00-14.45 Andrei Jaikin-Zapirain (Madrid)
15.00-15.45 David Conlon (Caltech)
16.15-17.00 Harald Helfgott (Goettingen)
Andrei Jaikin-Zapirain "Free Q-groups are residually torsion-free nilpotent".
A group G is called a Q-group if for any natural number n and any element g from G there exists a unique nth root of g in G. These groups were introduced by G. Baumslag in the sixties under the name of D-groups. The free Q-group on X can be constructed from the free group on X by applying an infinite number of amalgamations over cyclic subgroups. In this talk I will explain how to show that free Q-groups are residually torsion-free nilpotent. This solves a problem raised by G. Baumslag. A key ingredient of our argument is the proof of one instance of the Lueck approximation in characteristic p corresponding to an embedding of a finitely generated group into a free pro-p group.
For more details see http://matematicas.uam.es/~andrei.jaikin/preprints/baumslag.pdf.
http://matematicas.uam.es/~andrei.jaikin/preprints/slidesbaumslagparis.pdf
David Conlon, "Subset sums, completeness and colourings"
(joint work with Jacob Fox and Huy Tuan Pham.)
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive
integers, including solutions to several longstanding open problems. These include: solutions to the three problems of Burr and Erdos on Ramsey
complete sequences, for which Erdos later offered a combined total of $350; analogous results for the new notion of density complete sequences; the
solution to a conjecture of Alon and Erdos on the minimum number of colors needed to color the positive integers less than n so that n cannot be
written as a monochromatic sum; the exact determination of an extremal function introduced by Erdos and Graham on sets of integers avoiding a
given subset sum; and, answering a question of Tran, Vu and Wood, a strengthening of a seminal result of Szemeredi and Vu on long arithmetic
progressions in subset sums.
Harald Helfgott "Random walks on a divisibility graph"
(joint with Maksym Radziwiłł)
Let Γ be a graph having the integers N<n≤2N as its vertex set V, and (for all primes p in a range) an edge between n and n+p whenever p|n. We define an operator U on functions f:V→C in terms of the adjacency matrix of Gamma, and study its properties. We prove that, with few exceptions, the eigenvalues of U must all be small.
As one consequence, we establish a new bound for the logarithmic Chowla conjecture, viz.,
\[\sum_{n\leq x} \frac{\lambda(n) \lambda(n+1)}{n} = O\left(\frac{1}{(\log \log x)^c}\right)\]
for λ(n) the Liouville function and a fixed c>0. (Tao (2016) had proved a bound of o(1) by a different approach; his proof can be made to give a bound of O(1/log log log log x).)
Tuesday, May 26, in ZOOM,
Meeting ID: 854 0839 4187 Password: Consider a Cayley graph of a free group on 107 generators (with respect to this free generating set). What is the degree of this graph? Type the three digit number as the password.
.
15.00-15.45 Jacek Swiątkowski (Wrocław)
16.00-16.45 Corentin Le Coz (Orsay)
Jacek Swiątkowski, "Some results towards the topological classification of Gromov boundaries
of hyperbolic groups", Surprisingly little is still known about explicit topological spaces that can be realized as Gromov boundaries
of hyperbolic groups.I will discuss three results in this direction:
1. that all Gromov boundaries fall in the class of spaces called Markov compacta, constructed algorithmically
out of certain finite initial data (result of my student Dominika Pawlik);
2. that some particular Markov compacta - trees of manifolds and trees of graphs - appear among Gromov boundaries
of hyperbolic groups;
3. that Gromov boundaries of infinitely ended hyperbolic groups are explicitely understood in terms of Gromov
boundaries of 1-ended factors in their Stallings' decompositions.
Corentin Le Coz, "Separation and isoperimetric profiles",
The separation profile was defined for graphs by Benjamini, Schramm and Timar in 2011. It is a monotone coarse-geometric invariant, which means that it is able to give obstruction to the existence of a coarse embedding between graphs or spaces. In a joint work with Antoine Gournay, I have studied links between separation and isoperimetric profiles. I will present our results, with applications. This will lead us a new notion of local separation, that I will introduce at the end of the talk. I will give all the definitions and we will see many examples.
Tuesday, March 24 CANCELLED
14.00-14.45 Jacek Swiątkowski (Wrocław) POSTPONED
15.00-15.45 Corentin Le Coz (Orsay) POSTPONED
16.15- 17.00 Grigori Avramidi (Bonn)
Jacek Swiątkowski "Some results towards the topological classification of Gromov boundaries
of hyperbolic groups"
Surprisingly little is still known about explicit topological spaces that can be realized as Gromov boundaries
of hyperbolic groups.I will discuss three results in this direction:
1. that all Gromov boundaries fall in the class of spaces called Markov compacta, constructed algorithmically
out of certain finite initial data (result of my student Dominika Pawlik);
2. that some particular Markov compacta - trees of manifolds and trees of graphs - appear among Gromov boundaries
of hyperbolic groups;
3. that Gromov boundaries of infinitely ended hyperbolic groups are explicitely understood in terms of Gromov
boundaries of 1-ended factors in their Stallings' decompositions.
Corentin Le Coz, "Separation and isoperimetric profiles",
The separation profile was defined for graphs by Benjamini, Schramm and Timar in 2011. It is a monotone coarse-geometric invariant, which means that it is able to give obstruction to the existence of a coarse embedding between graphs or spaces. In a joint work with Antoine Gournay, I have studied links between separation and isoperimetric profiles. I will present our results, with applications. This will lead us a new notion of local separation, that I will introduce at the end of the talk. I will give all the definitions and we will see many examples.
Grigori Avramidi (TBA)
Tuesday, February 18,
room W, ENS, DMA
14.00-14.45 David Hume (Oxford)
15.00-15.45 Nikolay Bogachev (Skoltech & MIPT)
16.15- 17.00 Daniel Woodhouse (Oxford)
David Hume, "Three dichotomies for connected unimodular Lie groups".
Using the Levi decomposition theorem, Lie groups are usually studied in two separateclasses: semisimple and solvable. Both these classes further
divide into two subclasses with very different behaviour: semisimple groups split into the rank 1and higher rank cases; while solvable groups divide into those of polynomial growth and those of exponential growth. Amongst connected unimodular Lie groups, let us say that G is "small" if it shares a cocompact subgroup with some direct product of a rank one simple Lie group and a solvable Lie group with polynomial growth. Otherwise, we say G is "large". We present three strong dichotomies which distinguish "small" and "large" Lie groups; which are respectively algebraic, coarse geometric, and local analytic in nature. As an application we will show that Baumslag-Solitar groups admit a similar "small"/"large" dichotomy. This is part of a joint project with John Mackay and Romain Tessera.
Nikolay Bogachev, "Arithmetic hyperbolic reflection groups".
The story of hyperbolic reflection (Coxeter) groups and arithmetic reflection groups goes back to the 19th century, to the papers of
Poincare, Fricke, and Klein. Reflection groups on spheres and Euclidean affine spaces were classified by Coxeter in 1933. In 1967,
Vinberg developed the theory of hyperbolic reflection groups, where he proposed geometrical and combinatorial description of such groups, as
well as he proved the arithmeticity criterion for them. Due to the results of Vinberg (1967-1984), Nikulin (1980s, 2007), Agol,
Belolipetsky and other mathematicians (2005-2008) it is known that there are only finitely many maximal arithmetic hyperbolic reflection
groups in all dimensions. However, the problem of its classification (posed, in fact, by Vinberg in his papers 1967-1971) is still very far
from being completely solved. In this talk I will describe the known classification results and geometric approaches for solving this
problem, and also I will present my recent results on the classification of stably reflective hyperbolic lattices of rank 4 and
a software implementation of the Vinberg algorithm.
Daniel Woodhouse, "Action rigidity of free products of hyperbolic manifold groups".
Gromov's program for understanding finitely generated groups up to
their large scale geometry considers three possible relations: quasi-isometry, abstract commensurability, and acting geometrically on
the same proper geodesic metric space. A *common model geometry* for groups G and G' is a proper geodesic metric space on which G and G'
act geometrically. A group G is *action rigid* if any group G' that has a common model geometry with G is abstractly commensurable to G.
We show that free products of closed hyperbolic manifold groups are action rigid. As a corollary, we obtain torsion-free, Gromov
hyperbolic groups that are quasi-isometric, but do not even virtually act on the same proper geodesic metric space.
Tuesday, 21 January
room W, ENS, DMA
14.00-14.45 Francois Dahmani (Grenoble)
15.00-15.45 Tatiana Smirnova-Nagnibeda (Geneva)
16.15- 17.00 Urs Lang (ETH Zurich)
Francois Dahmani, "Relative hyperbolicity of free-by-cyclic groups, and conjugacy between automorphisms of free groups"
Given an automorphism of a free group, the associated semi-direct product is relatively hyperbolic with respect to mapping tori of its maximal polynomially growing subgroups. This statement of Gautero and Lustig recently received proofs of P. Ghosh, and of a joint work with R. Li, about semidirect products of free products and of Z. On the other hand, in some instances, due to joint works with V. Guirardel, and with N. Touikan, one can solve the isomorphism problem for relatively hyperbolic groups whose peripheral subgroups satisfy profinite separation properties. This opens possibilities regarding the algorithmic problem of conjugacy between two automorphisms of a free group, in favorable circumstances.
Tatiana Smirnova-Nagnibeda, "Various types of spectra and spectral measures on Schreier and Cayley graphs".
"Various types of spectra and spectral measures on Schreier and Cayley graphs" .We will be interested in the Laplacian on graphs associated with finitely generated groups: Cayley graphs and more generally Schreier graphs corresponding to some natural group actions. The spectrum of such an operator is a compact subset of the closed interval [-1,1], but not much more can be said about it in general.
We will discuss various techniques that allow to construct examples with different types of spectra: connected, union of two intervals, totally disconnected…, and how this depends on the choice of the generating set in the group. Types of spectral measures that can arise in these examples will also be discussed.
Urs Lang, "Characterizations and asymptotic geometry of rank-n-hyperbolic spaces".
In joint work with Bruce Kleiner we began a systematic study of spaces of higher asymptotic rank with a view towards higher dimensional analogues of (Gromov-)hyperbolic phenomena. In this talk I will discuss a number of different characterizations of such spaces as well as a rank n analogue of visual metrics at infinity, among others.
Tuesday, 17 December
room W, ENS, DMA
14.00-14.45 Urs Lang (ETH Zurich) CANCELLED, MOVED TO JANUARY 21
15.00-15.45 Ashot Minasyan (Southampton)
16.15-17.00 Assaf Naor (Princeton)
Urs Lang, "Characterizations and asymptotic geometry of rank-n-hyperbolic spaces"
In joint work with Bruce Kleiner we began a systematic study of spaces of higher asymptotic rank with a view towards higher dimensional analogues of (Gromov-)hyperbolic phenomena. In this talk I will discuss a number of different characterizations of such spaces as well as a rank n analogue of visual metrics at infinity, among others.
Ashot Minasyan, "CAT(0) groups with exotic properties"
During this talk I will discuss a class of groups that are higher-dimensional analogues of the classical Baumslag-Solitar groups. Many of their properties are also similar to the known properties of Baumslag-Solitar groups; e.g., they can be non-Hopfian and non-biautomatic. Nevertheless, some of these groups are CAT(0). The talk will be based on ajoint work with Ian Leary.
Assaf Naor, “Vertical-versus-horizontal isoperimetry"
We will describe an isoperimetric-type inequality in the Cayley graph of discrete Heisenberg groups which bounds from below the size of the edge boundary of any subset by its “vertical perimeter,” which is a quantitative measurement of the extent to which it is difficult to leave the set in the direction of the center. This inequality has applications to several areas, and it relies on a new multi-scale structural description of boundaries of sets in the Heisenberg group. It turns out that such vertical-versus-horizontal isoperimetry exhibits a marked difference between dimension 3 and dimension 5 (or higher). The goal of this talk is to present the aforementioned isoperimetric inequality, describe some of its applications, discuss the overall structure and ideas of its proof, and explain why there is a qualitative difference between dimension 3 and dimension 5. Based on joint work with Robert Young.
Tuesday, 12 November
room W, ENS, DMA
14.00-14.45 Peter Haissinsky (Université d'Aix-Marseille)
15.00-15.45 Nima Hoda (ENS Paris)
16.15-17.00 Elia Fioravanti (Oxford)
Peter Haissinsky, "Quasi-isometric rigidity of 3-manifold groups".
The talk will focus on the following results: a finitely generated group quasi-isometric to
the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group contains a finite index
subgroup isomorphic to the fundamental group of a compact 3–manifold or to a finitely generated Kleinian group.
This talk is based on joint work with Cyril Lecuire.
Nima Hoda, "Shortcut graphs and groups".
Shortcut graphs are graphs in which long enough cycles cannot embed without metric distortion. Shortcut groups are groups which act
properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in
geometric group theory and metric graph theory including: systolic and quadric groups (in particular finitely presented C(6) and C(4)-T(4)
small cancellation groups), cocompactly cubulated groups, hyperbolic groups, Coxeter groups and the Baumslag-Solitar group BS(1,2). Most of
these examples satisfy a strong form of the shortcut property. I will discuss some of these examples as well as some general constructions
and properties of shortcut graphs and groups.
Elia Fioravanti "Cross ratios on cube complexes and marked length-spectrum rigidity".
A well-known conjecture claims that the isometry type of a closed, negatively-curved Riemannian manifold should be uniquely determined by the lengths of its closed geodesics. By work of Otal, this is equivalent to the problem of extending cross-ratio preserving maps between Gromov boundaries of CAT(-1) manifolds. Progress on the conjecture has been remarkably slow, with only the 2-dimensional and locally symmetric cases having been completely solved so far. Nevertheless, it is natural to try leaving the world of manifolds and address the conjecture in the general context of non-positively curved metric spaces. We restrict to the class of CAT(0) cube complexes, as their geometry is both rich and well-understood. We introduce a new notion of cross ratio on their horoboundary and use it to provide an answer to the conjecture in this setting. More precisely, we show that essential, hyperplane-essential cubulations of Gromov-hyperbolic groups are completely determined by their combinatorial length functions. The same holds for possibly non-proper and non-cocompact actions of non-hyperbolic groups, as long as the cube complexes are irreducible and geodesically complete. Joint work (arXiv:1903.02447, arXiv:1810.08087) with J. Beyrer (Heidelberg).
Tuesday 29 October
room W, ENS, DMA
14.00-14.45 Christophe Pittet (Genève)
15.00-15.45 Thiebout Delabie (Orsay)
16.15-17.00 Alina Vdovina (Newcastle)
Christophe Pittet, "The exact convergence rate in the ergodic theorem of Lubotzky Phillips Sarnak".
We will show how to compute exact convergence rates in ergodic theorems with the help of Harish-Chandra's fonctions associated to boundary actions.
Thiebout Delabie, High dimensional cuts and coarse embedding.
The non existence of a coarse embedding of one finitely generated group into an other, is often established using some coarse property.
For example we know that a non-abelian free group does not embed coarsely into an abelian free group, since the former has exponential growth, while the later has polynomial growth.
However, if for both groups we take the direct sum with a non-abelian free group, then both groups have exponential growth.
In order to solve this issue, we will look at the separations profile and its generalisations.
Alina Vdovina, "Ramanujan cubical complexes as higher-dimensional expanders".
Ramanujan graphs were first considered by Margulis and Lubotzky, Phillips, Sarnak to get graphs with optimal spectral properties. In our days the theory of expander graphs
and, in particular, Ramanujan graphs is well developed, but the questions is what is the best definition of a higher-dimensional expander is still wide open. There are several approaches, suggested by Gromov, Lubotzky, Alon and others, but the cubical complexes were not much investigated from this point of view. In this talk I will give new explicit examples of cubical Ramanujan complexes and discuss possible developments.
2018-2019
Organized by Anna Erschler and Todor Tsankov
Supported by the ERC grant GroIsRan
Tuesday, 18 June
the talks will take place at the IHP, lecture room 1.
14.00-14.45 Katrin Tent (Münster)
15.00-15.45 Arindam Biswas (Vienna)
16.15-17.00 Laurent Bartholdi (Institut d'études avancées, ENS Lyon")
Katrin Tent, "Burnside groups of relatively small odd exponent"
(joint work with A. Atkarskaya and E. Rips)
Burnside asked whether a finitely generated periodic group is necessarily finite. While the question was
answered in the negative by Golod and Shafarevitch already in 1964, we consider the special case of finitely
generated groups of finite exponent. Adian and Novikov proved that the free Burnside group B(m,n), i.e. the
quotient of the free group F_m in m generators by the normal subgroup N generated by all nth powers is infinite for sufficiently large odd
exponent n.
We use special generators for the normal subgroup N and generalized small cancellation theory to improve the lower bound
on the exponent for which the free Burnside groups are infinite.
Arindam Biswas, "On minimal complements in groups"
Let $W, W'$ be non-empty subsets in an arbitrary group $G$. $W'$ is said to be a (right) complement to $W$ if $WW' = G$
and it is minimal if no proper subset of $W'$ is a complement to $W$. Minimal complements arise as group theoretic analgues
of minimal nets in metric geometry. In this talk we shall focus on several recent results on their existence and their inexistence.
Laurent Bartholdi, "Dimension series and homotopy groups of spheres"
The lower central series of a group GG is defined by γ_1=G and γ_n=[G,γ_{n−1}]. The "dimension series", introduced by Magnus, is defined using the group algebra over the integers: δ_n={g:g−1 belongs to the n-th power of the augmentation ideal}.
It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has δ_n≥γ_n, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with δ_4/γ_4 cyclic of order 2. On the positive side, Sjogren showed that δ_n/γ_n is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proven by Gupta) that only 2-torsion may occur.
In joint work with Roman Mikhailov, we prove however that for every prime p there is a group with p-torsion in some quotient δ_n/γ_n.
Even more interestingly, I will show that the dimension quotient δn/γnδn/γn is related to the difference between homotopy and homology: our construction is fundamentally based on the order-p element in the homotopy group π_2p(S^2) due to Serre.
Tuesday, 14 May
14.00-14.45 Stefan Witzel (Ecole Polytehchnique)
15.00-15.45 Katie Vokes (IHES)
15.45-16.15 coffee break
16.15-17.00 Adrien Le Boudec (ENS Lyon)
Stefan Witzel, "Arithmetic approximate groups and their finiteness properties"
I will talk about approximate groups, a geometric generalization of groups. Approximate groups were discovered
independently in various contexts and I will describe how they arise very naturally in the context of arithmetic groups. I will then
explain how to extend topological finiteness properties of groups to approximate groups. This allows to make a connection between
arithmetic groups in positive characteristic and arithmetic approximate groups in characteristic zero. The talk is based on joint
work with Tobias Hartnick.
Katie Vokes, "Hierarchical hyperbolicity of graphs associated to surfaces"
In a paper of 2000, Masur and Minsky studied the geometry of mapping class groups of surfaces using projections to certain Gromov hyperbolic graphs (the curve graphs) associated to subsurfaces. This inspired the definition by Behrstock, Hagen and Sisto of hierarchically hyperbolic spaces, which are equipped with similar projection maps, satisfying conditions which guarantee a structure analogous to that of the mapping class group. I will give some background on these concepts and present a result demonstrating that a large family of graphs associated to surfaces are hierarchically hyperbolic spaces.
Adrien Le Boudec, "Simple groups having a wreath product as a geometric model"
The goal of the talk will be to describe groups that are finitely
generated, simple, and that act properly and cocompactly on the
"natural" Cayley graph of the wreath product A \wr F, where A is a
finite group and F a non-abelian free group.
Tuesday, 16 avril
l'ENS, salle W (Toits du DMA)
14.00 -14.45 Cornelia Drutu (Oxford)
15.00- 15-45 Mathieu Dussaule (Nantes)
15.45-16.15 coffee break
16.15-17.00 Anthony Genevois (Orsay)
Cornelia Drutu "Fixed point properties and conformal dimension of the boundary for hyperbolic groups". Evidence from random group theory points out to the fact that for generic Gromov hyperbolic groups there is a connection between the conformal dimension of their boundary and properties related to affine actions by isometries on uniformly curved Banach spaces. In this talk I shall explain that there exist hyperbolic groups for which no such connection can be established. This is joint work with Ashot Minasyan
Mathieu Dussaule "The Guivarc'h inequality in relatively hyperbolic groups".
Consider a random walk on a finitely generated group. The Guivarc'h inequality states that h<lv, where h is the asymptotic entropy, l is the asymptotic drift and v is the volume growth of the group. We will be interested in the Guivarc'h inequality for finitely supported random walks on relatively hyperbolic groups. We will prove in particular that this is always a strict inequality when the maximal parabolic subgroups are virtually abelian.
Anthony Genevois
"Cubical geometry of braided Thompson's group". The braided Thompson group brV is a specific subgroup of the mapping class group of the plane minus a Cantor set which is constructed by imitating the definition of Thompson's group V as a group of homeomorphisms of the Cantor space. In this talk, I will explain how to construct an action of brV on an infinite-dimensional CAT(0) cube complex with cube-stabilisers isomorphic to braid groups, and how to deduce that polycyclic subgroups of brV are all virtually free abelian.
Tuesday, 12 mars
l'ENS, salle W (Toits du DMA)
14.00-14.45 Emmanuel Militon, (Nice)
15.00-15.45 Simon André, (Rennes)
15.45-16.15 coffee break
16.15-17.00 Nikolay Nikolov, (Oxford) [CANCELLED]
Emmanuel Militon, "Groups of diffeomorphisms of a Cantor set"
Let K be a Cantor set contained in the real line R. We call diffeomorphism of K a homeomorphism of K which is locally a restriction
of diffeomorphisms of R. In this talk, we will discuss some properties of the groups of diffeomorphisms of such Cantor sets and we will see
consequences of those results on Higman-Thompson's groups Vn.
Simon André, "Hyperbolicity is preserved under elementary
equivalence"
Zlil Sela proved ten years ago that any finitely generated group that satisfies the same first-order properties as a torsion-tree hyperbolic
group is itself torsion-free hyperbolic. This result is striking since hyperbolicity is defined in a purely geometric way. In this talk, I will
explain that Sela's theorem remains true for hyperbolic groups with torsion, as well as for subgroups of hyperbolic groups, and for
hyperbolic and cubulable groups.
Nikolay Nikolov, "On conjugacy classes in compact groups"
It's a classical result that the number of conjugacy classes of a finite group G tends to infinity as |G| tends to infinity. In this talk I will
present joint work with Andrei Jaikin-Zapirain. Our result is that an infinite compact group has uncountably many conjugacy classes. The proof
relies on the classification of finite simple groups and the studies of finite groups with almost regular automorphisms.
Tuesday, 12 February
l'ENS, salle W (Toits du DMA)
14.00-14.45 Joshua Frisch (Caltech)
15.00-15.45 Andy Zucker (Paris VII)
15.45-16.15 coffee break
16.15-17.00 Christophe Garban (Université Lyon 1)
Joshua Frisch, "Proximal actions, Strong amenability, and Infinite conjugacy class groups".
A topological dynamical system (i.e. a group acting by homeomorphisms on a compact topological space) is said to be proximal if for any two points p and q we can simultaneously push them together i.e. there is a sequence $g_n$ such that $lim g_n(p)=lim g_n (q)$. In his paper introducing the concept of proximality Glasner noted that whenever $\Z$ acts proximally that action will have a fixed point. He termed groups with this fixed point property "strongly amenable" and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent paper precisely characterizing which (countable) groups are strongly amenable. This work is joint with Omer Tamuz and Pooya Vahidi Ferdowski
Andy Zucker "Bernoulli Disjointness"
(Joint work with Eli Glasner, Todor Tsankov, and Benjamin Weiss)
We consider the concept of disjointness for topological dynamical
systems, introduced by Furstenberg. We show that for every discrete
group, every minimal flow is disjoint from the Bernoulli shift. We apply
this to give a negative answer to the “Ellis problem” for all such
groups. For countable groups, we show in addition that there exists a
continuum-sized family of mutually disjoint free minimal systems. In the
course of the proof, we also show that every countable ICC group admits
a free minimal proximal flow, answering a question of Frisch, Tamuz, and
Vahidi Ferdowsi.
Chrtistophe Garban, "Inverted orbits of exclusion processes, diffuse-extensive-amenability and (non-?)amenability of the interval exchanges"
After a brief introduction on the notion of amenability for groups, I will focus on the group of interval exchanges (the IET group) which is believed to be amenable. One of the (many) equivalent criteria to show that a group is amenable is Kesten's criterion on the return probabilities of random walks. In the case of G=IET, a recent work by Juschenko, Matte Bon, Monod and De La Salle provides a new criterion which is also of probabilistic nature. This new criterion involves the size of the inverted orbit of a certain random walk on the wobbling group W(Z^d) of permutations of Z^d. The aim of this talk will be to introduce natural models of random walks on permutations of Z^d for which this criterion can be analyzed. The talk will not require much prerequisites.
Tuesday, 22 January
14.00-14.45 Bertrand Remy (Ecole Polytechnique)
15.00-15.45 Tom Hutchcroft (Cambridge)
15.45-16.15 coffee break
16.15-17.00 Pavel Zalesski (University of Brasilia)
Bertrand Remy, "Quasi-isometric invariance of continuous group Lp-cohomology, and first applications to vanishings" (joint with Marc Bourdon)
We show that the continuous L^p-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M. Gromov, suggesting a classical behaviour of continuous L^p-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu’s vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings.
Tom Hutchcroft, "Kazhdan groups have cost 1"
I will sketch a proof that Kazhdan groups have cost 1, answering a question of Gaboriau. No knowledge of Kazhdan groups or of cost will be assumed. Joint work with Gabor Pete.
Pavel Zalesski, "The profinite completion of 3-manifold groups".
Abstract. We shall present structural results of the profinite completion \widehat G of a 3-manifold group G and its interrelation with the structure of G. We shall address the question to what extent the profinite completion of the fundamental group $\pi_1M of a 3-manifold determines the manifold M and discuss residual properties of \pi_1 M.
Tuesday, 11 December
14.00-14.45 Dawid Kielak (Bielefeld)
15.00-15.45 Henry Wilton (Cambridge) cancelled
15.45 -16.15 coffee break
16.15-17.00 Damian Osajda (Wroclaw and McGill, Montreal) 15.00-15.45
Dawid Kielak, "Fibring of residually finite rationally-solvable groups"
I will discuss how a little algebra can help in generalising results from low-dimensional topology to group theory. In particular, I will focus on a theorem of Ian Agol, who proved that if the fundamental group of a closed 3-manifold M satisfies the RFRS property, then M admits a finite covering which fibres over the circle. (Agol later showed that every hyperbolic 3-manifolds admits a finite covering whose fundamental group does satisfy the RFRS property, and thus completed the proof of Thurston's Virtually Fibred Conjecture).
In the talk I will indicate how to replace the 3-manifold by a finitely generated group G, thus obtaining an equivalence between G fibring (mapping onto the integers with a finitely generated kernel) and the vanishing of the first L^2 Betti number of G.
Henry Wilton, "Surface subgroups of cubulable hyperbolic groups" I’ll describe recent work which finds surface subgroups of certain graphs of free groups. When combined with the work Agol, Wise and others, it follows that all cubulable hyperbolic groups have surface subgroups, unless they are free. This answers a question of Gromov in this case. In particular, surface subgroups can be found in small-cancellation groups, and in hyperbolic mapping tori of free group automorphisms.
Damian Osajda, "A combination theorem for combinatorially non-positively curved complxes of hyperbolic groups"
Abstract: This is joint work with Alexandre Martin (Heriot-Watt University). Let X be a complex of hyperbolic groups. In general
the fundamental group of X need not to be hyperbolic. M. Bestvina and M. Feighn showed that if X is a graph of groups, and satisfies some
natural `acylindricity' conditions then the fundamental group of X is hyperbolic. A. Martin extended this combination theorem
to the case of X whose underlying complex carries a hyperbolic CAT(0) metric. I will present a combinatorial counterpart
of Martin's result obtained recently. We introduce a weak nonpositive-curvature-like combinatorial property and show that
fundamental groups of complexes of groups with underlying complex satisfying that property are hyperbolic. Our property
holds for e.g. (weakly) systolic complexes and small cancellation complexes giving rise to new examples of complexes of groups
with hyperbolic fundamental groups. The proof relies on constructing a potential Gromov boundary for the resulting groups
and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterization of hyperbolicity.
Tuesday, 13 November
14.00-14.45 Miklos Abert (Renyi Institute Budapest)
15.00-15.45 Arman Darbinyan (ENS Paris)
15.45-15.15 coffee break
16.15-17.00 Rachel Skipper (Göttingen and ENS Lyon)
Miklos Abert, "Kesten type theorems for groups, graphs and manifolds"
Kesten, in his influential PhD thesis proved that for a normal subgroup N of Gamma, the visiting exponent (or, spectral radius
of random walk) of N is greater than the visiting exponent of 1 if and only if N is amenable. In the talk we discuss generalizations of this
result in the realm of finite and infinite graphs, invariant random subgroups and Riemannian manifolds, in particular, locally symmetric
spaces. We also present some open problems. The results are due to Lyons-Peres, Gekhtman-Levit, Abert-Glasner-Virag, Mikolaj Fraczyk,
Mustazee Rahman and Abert-Bergeron-Virag.
Arman Darbinyan, "Word and conjugacy problems, and Tarski monsters"
The word and conjugacy problems for groups were introduced by Max Dehn in 1911 and are widely recognized as some of the most important features of finitely generated groups.
In the talk we discuss Tarski monsters (i.e. finitely generated non-cyclic groups with (finite) cyclic proper subgroups) with effective word and conjugacy problems.
We will also underline the method of such constructions and discuss some if its applications in wider context.
Rachel Skipper, "Finiteness properties of simple groups"
A group is said to be of type F_n if it admits a classifying space with compact n-skeleton. We will consider the class of Röver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type F_{n-1} but not F_n for each n. These are the first known examples for n >= 3.
As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups, the first is due to Caprace and Rémy.
This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky
Tuesday, 9 October
14.00-14.45 Martin Bridson (Oxford)
15.00-15.45 Arnaud Hilion (Marseille)
15.45-15.15 coffee break
16.15-17.00 Damien Gaboriau (ENS Lyon)
Martin Bridson, "Profinite rigidity and hyperbolic 3-orbifolds"
Developments in geometry and low dimensional topology have given renewed vigour to the following classical question: to what extent do the finite images of a finitely presented group determine the group? I'll survey what we know about this question in the context of 3-manifolds, and I shall present recent joint work with McReynolds, Reid and Spitler showing that the fundamental groups of certain hyperbolic orbifolds are distinguished from all other finitely generated groups by their finite quotients.
Arnaud Hilion, "Boundary of cyclic hyperbolic extensions of free groups".
Given a free group $F$ and an automorphism $\phi$ of $F$, we consider the mapping torus of $F$ by $\phi$. The resulting free-by-cyclic group $G$ is Gromov hyperbolic if and only if $\phi$ is atoroidal by the work of Brinkmann. When moreover the automorphism $\phi$ is fully irreducible, the work of Kapovich–Kleiner proves that the boundary of the group $G$ is homeomorphic to the Menger sponge. However, their proof is quite indirect, thus not giving tools to go further into the study of that boundary. I will explain how to construct an explicit embedding of a non planar graph in the boundary of $G$ whenever the group $G$ is hyperbolic (which in particular gives a new proof of the result of Kapovich-Kleiner). It is a joint work with Yael Algom Kfir and Emily Stark.
Damien Gaboriau, "On non-vanishing of the cohomology of Aut(F_n) and Out(F_n) in top dimensions".
Few results are know about the L^2-Betti numbers of Aut(F_n) and Out(F_n),
the groups of automorphisms (resp. outer automorphisms) of the free group F_n.
Their virtual geometric dimension (smallest dimension of a K(G,1) for torsion-free finite index subgroups) are 2n-2, resp. 2n-3.
I shall show that the top-dimensional L^2-Betti numbers of Aut(F_n) and Out(F_n) do not vanish.
By Lück approximation theorem, this implies that these groups admit finite index subgroups with non-vanishing top-dimensional
rational cohomology; in fact the usual Betti number for finite subgroups grows linearly with the index.
I will review the basics of the theory and stay at an elementary level.
2017-2018
Organized by Laurent Bartholdi and Anna Erschler
Tuesday, 26 June
IHP, salle 01
14.00-14.45 Pierre Pansu (Orsay)
15.00-15.45 Tianyi Zheng (San Diego)
15.45-15.15 coffee break
16.15-17.00 Sergei Ivanov (St-Petersbourg)
17.15-18.00 Kate Juschenko (Northwestern University of Chicago)
Pierre Pansu "L^1 cohomology of Euclidean spaces and Heisenberg groups"
It turns out that every closed L^1 differential form of degree <n on Euclidean n-space is the differential of an L^n/n-1 form. We shall explain why this is a bit unexpected, and why it fails for n-forms. Our arguments extend to Heisenberg groups.
Tianyi Zheng "Growth of torsion Grigorchuk groups"
On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. Joint with Anna Erschler.
Sergei Ivanov, "Transfinite invariants of groups and spaces"
Let R be one of the rings Z,Q,Z/p. Bousfield has defined the R-homological localization of a space X as the universal map in the homotopy category X --> X_R that induces an isomorphism on homology H_*(-,R). This notion is closely related to purely algebraic notion of HR-localization of a group, which can be defined as a version an algebraic closure with respect to some class of group equations. HR-localisation of a group is a transfinitely (R-)nilpotent group. We are interested in the length of the transfinite lower central series of HR-localization and call it HR-length of the group. For a free group F we proved that HZ-length(F)> omega+1; HZ/p-length(F)> omega; HQ-length(F)> omega.
We are also interested in HR-length of semidirect products of the cyclic group Z on Z^n, where the action of Z on Z^n is given by some matrix. We proved that under some assumptions the HZ-length of such group can be described on the language of eigenvalues of the matrix. Using this approach we show how to distinguish some fibrations (S^1)^n --> X --> S^1 up to homological equivalence. I will also show a connection of this theory with discrete homology of pro-p-groups and give a sketch of a prove that H_2(F,Z/p) is not trivial, where F is a free pro-p-group. The proof uses a lot of different techniques including Baire category theorem.
Kate Juschenko "On Liouville property of action of discrete groups"
We will discuss Liouville property of actions, relate it to amenability and to additive combinatorics for certain classes of groups. One of the central part of the discussion will be Thompson group F. We will discuss several open problems in additive combinatorics which relates to Liouville property of certain actions of Thompson group.
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Tuesday, 15 May
14.00-14.45 Thomas Schick (Göttingen University)
15.00-15.45 Jaques Darne (Lille)
15.45-15.15 coffee break
16.15-17.00 Volodymyr Nekrashevych (Texas A&M University)
Thomas Schick, "Approximation of L2-Betti numbers and the algebraic Eigenvalue property" (after
Jaikin-Zaipirain)
For a discrete group G and f an element of the rational group ring Q[G],
f acts as bounded operator on the Hilbert space of square summable functions on
G, extending the multiplication in C[G].
If G is finite, all eigenvalues of this operator are algebraic (being zeros of
the characteristic polynomial which has rational coefficients).
For infinite G, this is far less clear.
The main theme of the talk is to show that this, and the appropriate
generalizations to other coefficient subfields of C is true.
This is closely related to generalizations of the approximation of L2-Betti
numbers by their finite dimensional analogues (a theorem of Lück), and we will
explain these relations.
The general result is due to Andrei Jaikin-Zaipirain, with certain
simplifications of the special cases we want to address.
Jaques Darne, "Variations autour du problème d'Andreadakis"
Soit $F_n$ le groupe libre sur $n$ générateurs. On considère le groupe $IA_n$ des automorphismes de $F_n$ agissant trivialement sur son abélianisé. La structure de ce groupe est mal connue. Par exemple, si on lui connaît depuis longtemps un ensemble fini de générateurs explicites, on ne sait pas s'il est de présentation finie pour $n \geq 4$. Pour étudier sa structure, on peut définir deux filtrations ayant des propriétés similaires, l'une étant donnée par la structure interne du groupe, l'autre à partir de son action sur le groupe libre. Dans les années 1970, Andreadakis conjectura qu'elles coïncidaient. Néanmoins, de récents calculs ont infirmé cette conjecture. Dans cet exposé, je présenterai plusieurs variantes de ce problème, et quelques éléments de réponse.
Volodymyr Nekrashevych, "Amenability of iterated monodromy groups"
Amenability of the iterated monodromy groups of post-critically finite
rational functions is a wide-open problem. Another open problem is
amenability of groups generated by automata of polynomial activity
growth. In the talk, we will solve the intersection of these two open
problems. Namely, we will prove that if the iterated monodromy group
of a complex rational function is generated by an automaton of
polynomial activity growth, then it is amenable. This gives new
examples of amenable groups. By combining three well known theorems in
analysis, we will prove that the orbital graphs of the iterated
monodromy groups of rational functions are always recurrent, which
will imply amenability in the case of polynomially growing activity.
This is a joint work with D.Thurston and K.Pilgrim.
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Tuesday, 17 April
14.00 - 14.45 Omer Tamuz (Caltech)
15.00-15.45 Yair Hartman (Norwestern University)
15.45-16.15 coffee break
16.15-17.00 Said Sidki (University of Brasilia)
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Omer Tamuz, The Poisson boundary and the infinite conjugacy class property
Joint work with Joshua Frisch, Yair Hartman and Pooya Vahidi Ferdowsi.
The Poisson boundary of a random walk on a group captures the uncertainty in the walk's asymptotic behavior. It has long been known that all commutative groups are Choquet-Deny groups: namely, they have trivial Poisson boundaries for every random walk. More generally, it has been shown that all virtually nilpotent groups are Choquet-Deny. I will present a recent result showing that in the class of finitely generated groups, only virtually nilpotent groups are Choquet-Deny. This proves a conjecture of Kaimanovich and Vershik (1983), who suggested that groups of exponential growth are not Choquet-Deny. Our proof does not use the superpolynomial growth property of non-virtually nilpotent groups, but rather that they have have quotients with the infinite conjugacy class property (ICC). Indeed, we show that a countable discrete group is Choquet-Deny if and only if it has no ICC quotients.
Yair Hartman, Stationary C*-Dynamical Systems
Joint work with Mehrdad Kalantar
We introduce the notion of stationary actions in the context of C*-algebras, and prove a new characterization of C*-simplicity in terms of unique stationarity. This ergodic theoretical characterization provides an intrinsic understanding for the relation between C*-simplicity and the unique trace property, and provides a framework in which C*-simplicity and random walks interact.
Said Sidki, Self-similarity and finite presentability
We will review self-similarity and virtual endomorphisms of groups.
Then we follow with some recent results on the existence of lamplighters and with new constructions of finitely presented self-similar groups.
This is based on joints works with A. Dantas and D. Kochloukova.
Tuesday, 13 March
14.00-14.45 Julien Cassaigne (IML, Marseille)
15.00-15.45 Milton Minervino (LaBRI, Bordeaux)
15.45-16.15 coffee break
16.45-17.00 Nathalie Aubrun (ENS, Lyon)
Julien Cassaigne, A family of infinite words with complexity 2n+1 associated with a bidimensional continued fraction algorithm
After reviewing how Sturmian words and their s-adic expansions are linked with the usual additive continued fraction algorithm, we introduce a new continued fraction algorithm in two (projective) dimensions, that is designed to produce infinite words of complexity 2n+1. We study some of its properties and show that it is conjugated to an existing continued fraction algorithm, the Selmer algorithm. This is joint work with Sébastien Labbé (Bordeaux) and Julien Leroy (Liège)
Milton Minervino, Fractals de Rauzy et substitutions d'arbre
Les systèmes dynamiques symboliques substitutifs dits de type Pisot peuvent être interprétés géométriquement par des fractals introduits pour la première fois par G. Rauzy.
La célèbre conjecture de Pisot affirme que ces systèmes ont spectre discret et cela revient à montrer que les fractals de Rauzy pavent l'espace où ils sont représentés.
Le but de cet exposé sera de montrer comment construire des arbres auto-similaires qui remplissent ces fractals, lorsque les substitutions considérées soient aussi des automorphismes du groupe libre paragéométriques. Cela permet de relier la dynamique du système substitutif avec un échange d'intervalles.
Nathalie Aubrun, Tilings problems on substitution orbits
Given a substitution on words, we can associate to every orbit of a bi-infinite word a tiling of R^2 with hyperbolic tiles. In this talk I will explain how we can superimpose two orbits from two different substitutions and obtain an orbit of a non-deterministic substitution. We then use this construction to prove the undecidability of the Domino problem on surface groups. This is joint work with Etienne Moutot and Sebastian Barbieri.
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Tuesday, 20 February
Salle W
14.00-14.45 François Ledrappier (Paris VI)
15.00-15.45 Johannes Cuno (ENS)
15.45-16.15 coffee break
16.15-17.00 Ariel Yadin (Ben Gurion University)
François Ledrappier: Differentiability of the linear drift in negative curvature
We consider the linear drift of the Brownian motion on the universal cover of a compact manifold with negative curvature. We are interested in the differentiability of this drift as a function of the metric. For a one-parameter $C^k$ family of $C^k$ metrics with negative curvature, the drift is $C^{k-3}$. This is a joint work with Lin Shu (Peking U).
Johannes Cuno, Random walks on Baumslag-Solitar groups
We discuss how Kaimanovich's strip criterion can be used to identify the Poisson boundary of random walks on Baumslag-Solitar groups. This is is based on a joint publication with Ecaterina Sava-Huss. Time permitting, we conlclude witha few remarks concerning the question whther the asymptotic entropy depends
continuously on the measure driving the random walk.
Ariel Yadin, Intersectional IRS and Furstenberg entropy realization
Given a random walk on a (finitely generated) group the limiting entropy is a very useful quantity, with many applications. Perhaps most famously it has been linked to the existence of bounded harmonic functions (Poisson boundary): the limiting entropy is positive if and only if there exist non-constant bounded harmonic functions.
Given a subgroup, one may wish to consider the quotient group, and consider the entropy of the projected random walk. For a non-normal subgroup this does not necessarily converge. However, for an invariant random subgroup (IRS) the limiting entropy always exists and is bounded between 0 and the group's original limiting entropy, h.
We study the question: which values in the interval [0,h] can be realized by quotients of the group.
Along the way we provide an original construction on IRSs under some general algebraic conditions.
As a result, we are able to analyze lamp-lighter groups and other similar semi-direct products quite simply, and show that the full interval of entropies may be realized.
Moreover, using a geometric construction on the free group we are able to extend results of Bowen. We show that for any finitely supported random walk on the free group the full interval of entropies may be realized.
All notions such as IRSs and entropy will be explained in the talk.
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Tuesday, 16 January
Salle W
14.00-14.45 Nicolas Matte Bon (ETH Zürich)
15.00-15.45 Alejandra Garrido (Düsseldorf)
15.00-16.15 coffee break
16.15-17.00 Francois Le Maître (Paris VII)
Nicolas Matte Bon, Graphs of germs and homomorphisms of full groups
To any group or pseudogroup of homeomorphisms of the Cantor set one can associate a larger (countable) group, called the topological full group. It is a complete invariant of the groupoid of germs of the underlying action (every isomorphism between full groups is implemented by a conjugacy of the corresponding pseudogroups). I will discuss a theorem on the possible actions on topological full groups on compact spaces, which can be seen as a generalisation of this fact and has various new consequences.
Alejandra Garrido , Profinite completions of groups acting on rooted trees
Groups of rooted tree automorphisms, and (weakly) branch groups in particular, have received considerable attention in the last few decades, due to the examples with unexpected properties that they provide, and their connections to dynamics and automata theory.
These groups also showcase interesting phenomena in profinite group theory. I will discuss some of these and other profinite completions that one can use to study these groups, and how to find them.
Francois Le Maître, L1 full groups and entropy
L1 full groups are a measurable analogue of topological full groups of homeomorphisms on the Cantor space. They are complete invariants of flip-conjugacy for measure-preserving ergodic transformations by a combination of results by Belinskaya and Fremlin, so their algebraic properties should completely reflect the properties of the transformation. Unfortunately, unlike their topological siblings, they are uncountable, which makes their study as abstract groups difficult.
The good news is that they still carry a Polish group topology. They moreover have a well defined quasi-isometry type, although we still don’t know how to distinguish two L1 full groups up to quasi-isometry. In this talk, we will focus on the existence of dense finitely generated groups in the L1 full group of an ergodic transformation, and we will see that this happens if and only if the transformation has finite entropy.
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Tuesday, 5 December
Salle W
14.00-14.45 Markus Steenbock (ENS)
15.00-15.45 Dominik Francoeur (ENS)
15.45-16.15 coffee break
16.15-17.00 Feyishayo Olukoya (St-Andrews)
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Markus Steenbock "Product Set Growth in Hyperbolic Geometry".
(joint with Thomas Delzant). Improving a theorem of Razborov (Ann. of Math. 2014), Safin (Mat. Sb.
2011) proved that there is a positive constant c such that for every finite subset U of a free group that does not generate a cyclic subgroup
|U^3| > c |U|^2.
We will discuss the extension of this result to hyperbolic groups. This includes cocompact Fuchsian groups, fundamental groups of compact
Riemannian manifolds of negative sectional curvature, and finitely presented small cancellation groups.
Dominik Francoeur
"On maximal subgroups in branch groups". Branch groups are a class of groups acting on rooted trees in a way that mimics in some aspects the action of the whole group of automorphisms of the tree. They are a rich source of examples of groups with interesting properties, such as finitely generated infinite torsion groups and groups of intermediate growth. The study of maximal subgroups in branch groups was initiated by E. Pervova in 2005, when she showed that for the torsion Grigorchuk groups and many other related groups, every maximal subgroup is of finite index. In this talk, I will show that this result does not generalize to all branch groups of intermediate growth, or even to all Grigorchuk groups.
Feyishayo Olukoya "The growth rates of groups generated by reset automata".
Groups generated by automata (automata groups) are an interesting class of groups and have furnished counterexamples to several conjectures. In this talk we study groups generated by reset automata. Such groups have been studied by Silva and Steinberg who show that they are either finite, or locally-finite-by-infinite cyclic and give sufficient conditions for when these groups have exponential growth rate. Using a different method we demonstrate that a group generated by a reset automata is either finite or has exponential growth rate extending the results of Silva and Steinberg. This is in the same vein as a recent result by Ines Klimann for the class of bireversible Mealy automata, however, we obtain our results using different techniques.
Tuesday, 14 November
Salle W
14.00-14.45 Roman Mikhailov (St.Petersburg) "Around nilpotent completion".
15.00-15.45 Gilbert Levitt (Caen) "On elementary equivalence of hyperbolic groups".
15.45-16.15 coffee break
16.15-17.00 Ivan Mitrofanov (ENS) Algorithmic problems for self-similar groups.
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Roman Mikhailov "Around nilpotent completion" .
The talk is based on two recent papers written jointly with S. Ivanov: https://arxiv.org/abs/1705.09131, https://arxiv.org/abs/1708.00282
For a free noncyclic group F, the pro-p-completion of F has uncountable homology group H_2 with Z/p-coefficients. This answers a problem of A. K. Bousfield. Analogous result holds for the case of Q-completion and rational H_2. This implies that, a wegde of circles is a Q-bad space in the sense of Bousfield-Kan.
Gilbert Levitt, "On elementary equivalence of hyperbolic groups".
I will discuss various aspects of the classification of hyperboli groups up to first-order equivalence
Ivan Mitrofanov "Algorithmic problems for self-similar groups".
Self-similar groups are generalization of automata groups: a faithful action of a group G on A* is called self-similar if for every g in G and a in A there exist h in G and b in A such that g(aw) = bh(w) for all words w in A*. We also prove that the word problem is undecidable in finitely generated self-similar groups.
Tuesday, 17 October,
14.00-14.45 Jérémie Brieussel (Montpellier)
15.00-15.45 Adrien Boyer (Paris VII)
15.45-16.15 coffee break
16.15-17.00 Leonid Potyagailo (Lille)
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Jérémie Brieussel, "Speed of random walks in finitely generated groups", The speed of a random walk is the average distance from the starting point as a function of time. Given an arbitrary (regular) function between diffusive and linear, we construct a group (and a probability) with this speed function up to multiplicative constant. The isoperimetric profile and the L_p-compression of this group can also be computed. This is a joint work with Tianyi Zheng.
Adrien Boyer, "A new proof of Property RD for hyperbolic groups via the boundary". I will give a new proof Property RD satisfied by hyperbolic groups. This result goes back to Jolissaint and de la Harpe.
The proof that I will discuss uses boundary representations. The main point of these new techniques, based on ergodic geometry, is to give a possible new approach to Valette’s conjecture asserting that compact lattices in higher rank should satisfy property RD.
Leonid Potyagailo, "Martin boundary covers Floyd boundary". This is a joint work with I. Gekhtman (Yale, USA), V. Guerassimov (Belo Horisonté, Brazil) and Wenyuan Yang (Bejing, China).
We consider a random walk on the Cayley graph of a finitely generated group corresponding to a symmetric probability measure whose support generates the group. The Martin compactification is the unique minimal compactification of the graph to which all Martin kernels extend continuously. The Martin boundary is the remainder of this compactification.
Another compactification is the Floyd compactification which is the Cauchy completion of the Cayley graph equipped with the metric obtained by the rescaling of the word metric by a suitable real positive function. The Floyd boundary is the remainder of this compactification.
One of our central results claims that the identity map from the group to itself extends to a continuous equivariant surjective map from the Martin compactification to the Floyd compactification. Furthermore the preimage of every conical point
at the Floyd boundary is a single point at the Martin boundary. As a corollary we obtain that there exists an equivariant continuous and surjective map from the Martin boundary to the Bowditch boundary of a relatively hyperbolic group; moreover this map is injective on the preimage of the set of conical points whose complementary set is a countable set of parabolic points.
The proofs of these results essentially use an inequality which relates the Floyd and Green metrics and which generalizes the Ancona inequality for hyperbolic groups.
2016-2017
Tuesday, 13 June, 14.00-17.00
Volodymyr Nekrashevych
Rémi Coulon
This is a joint work with L.Bartholdi.
There is no algorithm for determining whether a given finite-state transducer has finite order. We prove this by embedding Minsky machine computations into automata groups.
Cyril Houdayer
Volodymyr Nekrashevych (TAMU) "Inverted orbits and simple groups of intermediate growth"
We will talk about a new construction of torsion groups:
fragmentations of minimal non-free actions of the infinite dihedral
group. When the fragmentation satisfies some low complexity condition,
then the group is of intermediate growth. One can construct first
examples of simple groups of intermediate growth in this way. The main
technique for the proof of sub-exponential growth are inverted orbits,
as developed by L. Bartholdi and A. Erschler. We will also discuss
open questions.
Rémi Coulon (Rennes) "On the variety of Burnside groups"
(joint work with D. Gruber)
The variety of Burnside groups of exponent n, denoted by B(n), is the class of groups satisfying the law x^n = 1. In 1902, W. Burnside asked whether all finitely generated groups in B(n) are finite. It turns out that the answer is negative provided n is sufficiently large (Novikov-Adian, Ol'shankskii, Lysenok, Ivanov). Actually B(n) is a very rich class of groups. Although they are torsion groups, free Burnside groups have at a local scale some "hyperbolic features". In this talk we will explain how one can take advantage of this idea to design Burnside groups with prescribed properties. This approach combines small cancellation theory and acylindrical action on hyperbolic spaces. Its strength is that it can be used without any knowledge on Burnside groups. As an application we recover some known result such as the SQ-universality of free Burnside groups. We also use this method to produce new examples of groups (e.g. a Gromov monster with bounded torsion) or show the unsolvability of numerous decision problems in B(n).
Cyril Houdayer (Orsay) "Strong ergodicity vs. spectral gap for group actions on measure spaces"
It is well-known that for probability measure preserving (pmp) group actions, if the associated Koopman representation has spectral gap (i.e. has no almost invariant vectors), then the action is strongly ergodic (i.e. has no non-trivial almost invariant measurable subsets). The converse is however not true as demonstrated by Schmidt’s example.
In this talk, I will first present a characterization of strong ergodicity for arbitrary nonsingular group actions in terms of spectral gap of the full group of the associated orbit equivalence relation. I will then explain how this criterion can be used to characterize strong ergodicity of the Maharam extension of group actions of type III in terms of a new invariant, analogous to Connes \tau invariant for type III factors. I will finally show that for strongly ergodic free actions of free groups, Connes’ \tau invariants of the action and of the associated group measure space factor coincide. This is joint work with A. Marrakchi and P. Verraedt.
Tuesday, 23 May, 14.00-17.00
Khalid Bou-Rabee,
Vadim Kaimanoich,
Yves de Cornulier
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Khalid Bou-Rabee (CUNY) "The Primitive Burnside Problem"
Let P(a,k) be the subgroup of the rank a free group generated by kth powers of primitive elements. We show that P(2,k) is finite index if and only if k=1 or 2 or 3. We frame this as a solution to the Primitive Burnside Problem and discuss applications to the Bounded Burnside Problem. This covers joint work with Patrick W. Hooper
Vadim Kaimanovich (Ottawa) "L-amenable actions"
The notion of amenability was first extended from groups to transitive group actions by Greenleaf (1969) by requiring that there exist an invariant mean on the action space. However, shortly thereafter this notion was eclipsed by Zimmer's definition (which is in a sense complementary to Greenleaf's one) and remained almost forgotten until fairly recently.
Amenability of a transitive action of a finitely generated group is equivalent to amenability of the associated Schreier graph, and the growing interest in properties of Schreier graphs has made Greenleaf's definition popular again during the last decade.
Inspired by a recent paper by Juschenko and Zheng on the Liouville property for Schreier graphs, I will introduce yet another version of amenability for actions and discuss its basic properties.
Yves de Cornulier (Orsay) "QI and SBE classification of nilpotent groups"
The starting point of this talk is the following conjecture: two simply connected nilpotent Lie groups are quasi-isometric (QI) if and only if they are isomorphic. I will describe the main known results in this direction: Pansu's theorem (QI-invariance of the associated Carnot graded Lie algebra) and Shalom-Sauer's theorems (QI-invariance of the cohomology algebra). I will also discuss the SBE-classification. SBE (Sublinearly Bilipschitz Equivalence) are defined in the same way as QIs, but are a weakening since bounded terms are replaced by sublinear error terms in the definition. Pansu's theorems provide a full SBE classification of these nilpotent Lie groups (and thus of finitely generated ones) in terms of graded Lie algebras (by the way, SBEs provide a way to state them without reference to asymptotic cones). Nevertheless, when we look in a quantitative way, i.e. we try to quantify the rate of sublinear growth of error terms in SBEs, we obtain natural refinements of the quasi-isometry classification. I will explain this in detail and provide partial results.
Tuesday, 4 April, 14-00-17.00, DMA, Salle W, toits du DMA
14.00.-14.45 Bruno Duchesne
15.00-15.45 Emmanuel Breuillard
15.45-16.15 coffee break
16.15-17.00 Henry Bradford
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Bruno Duchesne (Université de Lorraine) "Groupes d’homéomorphismes de dendrites"
Résumé : Les dendrites sont objets topologiques simples : elles sont par définition compactes, connexes, localement connexes, métrisables et chaque paire de points est reliée par un unique arc.
Ces dendrites sont très proches des arbres réels à la différence que l’on ne choisit pas de distance. Le cadre n’est pas métrique mais purement topologique.
Le but de l’exposé serait de décrire des propriétés structurelles et topologiques des groupes d’homéomorphismes de dendrites. Par certains aspects, ces propriétés sont réminiscentes des groupes agissant sur les arbres et radicalement opposées par d’autres aspects. Il s’agit d’un travail en commun avec Nicolas Monod.
Emmanuel Breuillard (Münster) "Free subgroups in linear groups"
abstract: The Tits alternative asserts that every non virtually solvable subgroup of GL(d,C) is contains a non abelian free subgroup. It is interesting to find free subgroups with extra properties. For example can one find a free subgroup of SO(2n) acting freely on the (2n-1)-dimensional sphere ? This question has been answered positively by Deligne and Sullivan in the 80's using certain division algebras. In connection to this is the celebrated Banach-Tarski paradox, and more precisely the minimal number of pieces in a paradoxical decomposition of a set on which a group acts. We will formulate a conjecture to the effect that every algebraic action of a linear group on an algebraic variety, that is paradoxical, is paradoxical with at most 4 pieces. We will then show that it is equivalent to a certain refinement of the Tits alternative, and prove it in some important cases. Joint work with B. Guralnick and M. Larsen.
Henry Bradford, (Göttingen) "Short Laws for Finite Groups and Residual Finiteness Growth"
Abstract: A law for a group G is a non-trivial word which is a relation between all pairs of elements in G. We study the asymptotic growth of the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.
Tuesday, 21 March , 14-00-17.00, DMA, Salle W, toits du DMA
14.00-14.45 Camille Horbez "Boundary amenability of Out(Fn)"
15.00-15.45 Romain Tessera "Poincaré profile in Hyperbolic groups"
15.45-16.15 coffee break
16.15-17.00 Yash Lodha "Nonamenable groups of piecewise projective homeomorphisms"
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Camille Horbez (Orsay), "Boundary amenability of Out(Fn)",
Abstract: This is joint work with Mladen Bestvina and Vincent Guirardel. We prove that the group Out(Fn) of outer automorphisms of a finitely generated free group is boundary amenable, i.e. it admits a topologically amenable action on a compact Hausdorff space. This also holds more generally for Out(G), where G is either a Gromov hyperbolic group or a right-angled Artin group. This implies in particular that Out(Fn) (and all these groups Out(G)) satisfies the Novikov conjecture on higher signatures. The strategy of our proof is partly inspired by Kida's proof of the analogous result for mapping class groups of finite type surfaces (also due to Hamenstädt).
Romain Tessera (Orsay), "Poincaré profile in Hyperbolic groups"
Abstract: In a joint work with David Hume and John Mackay, we define a family of coarse analytic invariants for graphs (and more generally metric measure spaces) that generalize the separation profile, introduced by Benjamini and Schramm. We calculate these invariants for groups with polynomial growth, the real hyperbolic spaces and for Bourdon's buildings. We also obtain estimates for other symmetric spaces. These estimates allow us for instance to prove that the complex hyperbolic plane does not coarsely embed into the real hyperbolic space of dimension 4.
Yash Lodha (EPFL Lausanne), "Nonamenable groups of piecewise projective homeomorphisms"
Abstract: Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free
subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure and
Tarski numbers.
Tuesday, 21 February , 14-00-18.00, DMA, Salle W, toits du DMA
Francois Dahmani (Rennes)
John MacKay (Bristol)
Vincent Guirardel (Grenoble)
Shahar Mozes (Hebrew University of Jerusalem)
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Francois Dahmani "Projection complexes and rotating families: analysing the normal closure of a Dehn twist".
John MacKay "A coarser contracting boundary"
Abstract: In a Gromov hyperbolic space, geodesics have the property of being contracting. Recently Charney and Sultan used this idea to define a new boundary of CAT(0) spaces which is invariant under quasi-isometries. I will discuss work with Christopher Cashen where we define a new topology on the space of contracting geodesics in any finitely generated group, and how parts of the theory of hyperbolic boundaries generalises to this context.
Vincent Guirardel "Endomorphisms of lacunary hyperbolic groups".
By definition, a lacunary hyperbolic groups is a finitely generated group having one asymptotic cone that in an R-tree.
This class of groups contains many wild groups such as infinite torsion groups, groups having no proper non-cyclic subgroups, or groups with a properly embedded expander.
However, we show that there are strong constraints on the set of endomorphisms of such a group G.
Indeed, we prove that any lacunary hyperbolic group is Hopfian (any surjective endomorphism is an automorphism).
In addition, we show that if a lacunary hyperbolic group has the fix point property for actions on $\mathbf R$-trees, then it is co-Hopfian (any injective endomorphism is an automorphism) and its outer automorphism group is locally finite.
We also construct lacunary hyperbolic groups whose automorphism group is infinite, locally finite, and contains any locally finite group given in advance. This is a joint work with Rémi Coulon.
Shahar Mozes "Topological finite generation of certain compact open subgroups of tree automorphisms".
Abstract: Given a finite permutation group $F<Sym(d)$ one can define a group of automorphism of a $d$-regular tree whose local action on the tree is given by the permutation group $F$. In a joint work with Marc Burger we determine when the maximal compact subgroup of this group is topologically finitely generated. This is motivated by studying uniform lattices in the group of automorphisms of a product of trees.
Tuesday, 24 january, DMA, Salle W, toits du DMA
"Property T and fixed point properties"
14.00-14.45 Marc Bourdon
15.00-15.45 Masato Mimura
15.45-16.15 coffee break
16.15-17.00 Mikael de la Salle
Marc Bourdon (Lille) "Espaces hyperboliques, dimension conforme et cohomologie $\ell _p$".
Résumé : La dimension conforme du bord d'un espace hyperbolique (au sens de Gromov) est un invariant numérique de quasi-isométrie de l'espace.
Elle est minorée par la dimension topologique du bord et majorée par sa dimension de Hausdorff. Dans cet exposé on présentera
la dimension conforme, et on s'intéressera à déterminer les groupes hyperboliques dont la dimension conforme du bord est égale à la dimension
topologique. Pour cela on utilisera des liens entre la dimension conforme et la cohomologie $\ell _p$.
Masato Mimura (EPFL Lausanne) "Superintrinsic synthesis in fixed point properties".
Abstract: The following natural question arises from Shalom's innovational work (1999, Publ.IHES) on Kazhdan's property (T). ``Can we establish an `intrinsic' criterion to synthesize relative fixed point properties into the whole fixed point property without assuming `Bounded Generation'?'' This talk is aimed to present a resolution to this question in the affirmative. Our criterion, moreover, suggests a further step toward constructing super-expanders from finite simple groups of Lie type.
In the talk, precise meanings of being `intrinsic' and `Bounded Generation' will be described. No deep background on the concerned topics is required.
Mikael De La Salle (ENS Lyon), "Zuk's criterion for actions on Banach spaces".
Abstract: Zuk's criterion for property (T) states that if a finitely generated group admits a Cayley graph whose links have spectral gap $> \frac 1 2$, then every action of it by isometries on a Hilbert space has a fixed point. I will explain that the same holds in one replaces Hilbert space by "uniformly curved Banach space" (for example $L_p$ space in the reflexivity range) and $>\frac 1 2$ by $1-\varepsilon$. This applies to random groups in Gromov's density $>\frac 1 3$ model. Based on a joint work with Tim de Laat.
Tuesday, 8 december
Afternoon in holomorphic dynamics, http://www.math.ens.fr/~laurentb/
Tuesday, 8 november, DMA/ENS, Salle INFO 2, bâtiment Rataud
"Random walks on groups"
14.00-14.45 Tianyi Zheng (Stanford) "Speed of random walks and minimal growth of harmonic functions on groups".
15.00-15.45 Antoine Gournay (Koeln) "Mixing, malnormal subgroups and cohomology in degree one".
15.45-16.15 coffee break
16.15-17.00 Sébastien Gouëzel (Nantes) "Numerical estimates for the spectral radius in surface groups".
Abstracts:
Tianyi Zheng "Speed of random walks and minimal growth of harmonic functions on groups".
We discuss a construction of groups where the speed (rate of escape) of simple random walk can follow any sufficiently regular function between diffusive and linear. When the speed of the \mu-random walk is sub-linear, all bounded \mu-harmonic functions are constant. We investigate the minimal growth of non-constant harmonic functions on these groups and show it is tightly related to the speed of the random walk.
Based on joint works with Jeremie Brieussel, Gidi Amir and Gady Kozma.
Antoine Gournay (Koeln) "Mixing, malnormal subgroups and cohomology in degree one".
Sébastien Gouëzel "Numerical estimates for the spectral radius in surface groups".
Estimating numerically the spectral radius of a random walk on a nonamenable graph is complicated, since the cardinality of balls grows exponentially fast with the radius. We propose an algorithm to get a bound from below for this spectral radius in Cayley graphs with finitely many cone types (including for instance hyperbolic groups), and discuss in particular the case of surface groups.
2015-2016
Alexei Kanel-Belov (Universite Bar Ilan et MIPT, Moscou),
"On the geometric ring theory" mardi, 3 mai, 14.00 a l'ENS, salle U-V, 2ème sous sol de DMA.
Abstract:
Quite recently Eliyahu Rips and Arye Juhasz constructed an Engel but not locally nilpotent group, i.e. group which satisfies for some positive $n$ the identity $\underbrace{[x,y],y,\dots,y]\dots]}_n=e$.
This group has non-postitive curvature and big commutative parts, some parts have small cancellation and some commute. - This group looks in some sense like a ring, and group multiplication behaves sometimes like multiplication and sometimes like addition. The theory of canonic forms of this group is applicable for rings, in particulary in skew field construction.
In different sense some semigroup constructions can be transformed to rings.
There is a hope nowdays to develop a geometric ring theory.
Narutaka Ozawa (Kyoto), "A functional analysis proof of Gromov's polynomial growth theorem" :
mardi, 5 avril, 14.00-15.00, l'ENS, salle W (Toits du DMA).
Abstract: The celebrated theorem of Gromov in 1980 asserts that any finitely generated group with polynomial growth is virtually nilpotent, i.e., it contains a nilpotent subgroup of finite index. Alternative proofs have been given by Kleiner (2007), etc. In this talk, I will give yet another proof of Gromov's theorem, based on functional analysis and random walk techniques.