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,
Videos of the talks
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
