Asymptotic properties of groups March 24  28, 2014 Organizers: Emmanuel Breuillard, Indira Chatterji and Anna Erschler Videos Conference photo (taken by Evgeny Plotkin) Invited speakers:
Program for the week: Monday March 24 (amphi Friedl) 09:45  10:15 COFFEE and REGISTRATION at IHP ground floor 10:15  11:00 Rostislav Grigorchuk: Invariant random subgroups of groups of the lamplighter type and some geometric groups. 11:15  12:00 Martin Bridson: Profinite isomorphism problems LUNCH BREAK 14:00  14:45 Elyahu Rips: Free Engel Groups and Similar Groups. 15:00  15:45 Yves Benoist: Random walk on padic flag varieties. 15:45  16:15 COFFEE BREAK 16:15  17:00 Nikolay Nikolov: Right angled cocompact lattices in higher rank simple Lie groups. Tuesday March 25 (amphi Hermitte) 10:00  10:45 Elon Lindenstrauss: Spectral gap, random walks on Euclidean isometry groups, and selfsimilar measures. 10:45  11:15 COFFEE BREAK 11:15  12:00 Thomas Delzant: Holomorphic families of Riemann surfaces from the point of view of asymptotic group theory. LUNCH BREAK 14:00  14:45 Alexander Olshanskii: Relative growth of subgroups in finitely generated groups. 15:00  15:45 POSTER SESSION 15:45  16:15 COFFEE BREAK 16:15  17:00 Laurent Bartholdi: Imbeddings in groups of subexponential growth. Wednesday March 26 (amphi Hermitte) 10:00  10:45 Alex Lubotzky: Quantum error correcting codes and asymptotic properties of 4dimensional arithmetic hyperbolic manifolds. 10:45  11:15 COFFEE BREAK 11:15  12:00 Joel Friedman: Sheaves on Graphs, L^2 Betti Numbers, and Applications. 12:15  13:00 Nicolas Monod: Cutting and pasting: a group for Frankenstein. FREE AFTERNOON Thursday March 27 (amphi Hermitte) 10:00  10:45 Amos Nevo: On best possible rates of Diophantine approximation by lattice orbits. 10:45  11:15 COFFEE BREAK 11:15  12:00 Zlil Sela: Envelopes and equivalence relations in a free group. LUNCH BREAK 14:00  14:45 Mladen Bestvina: On the asymptotic dimension of a curve complex. 15:00  15:45 John S. Wilson: Metric ultraproducts of finite simple groups. 15:45  16:15 COFFEE BREAK 16:15  17:00 Mark Sapir: The Tarski numbers of groups. 18:30 : COCKTAILDINNER in Jussieu (participants who registered for this cocktail will receive the instructions per email) Friday March 28 (amphi Hermitte) 10:00  10:45 Miklos Abert: Invariant random subgroups. 10:45  11:15 COFFEE BREAK 11:15  12:00 Leonid Potyagailo: Similar Relatively hyperbolic actions of a group. LUNCH BREAK 14:00  14:45 Gilbert Levitt: Vertex finiteness for relatively hyperbolic groups. 15:00  15:45 Karen Vogtmann: Outer space for rightangled Artin groups. 15:45  16:16 COFFEE and END OF CONFERENCE If you want to make a poster
presentation, please email your name, affiliation, title and abstract. Abstracts: Miklos Abert (Alfred Renyi Institute, Hungary) Title: Invariant random subgroups. Abstract: An invariant random subgroup (IRS) of a group is a random subgroup whose distribution is invariant under the conjugation action of the ambient group. IRSes tend to behave like normal subgroups in the sense that results that hold for normal subgroups but not for arbitrary subgroups tend to generalize to IRS's. Also, weak convergence of IRS's translates to BenjaminiSchramm convergence of the corresponding quotient spaces. These phenomena can be exploited in various ways. In the talk I will survey the known results and directions and pose some questions. Laurent Bartholdi (University of Gottingen, Germany) Title: Imbeddings in groups of subexponential growth Abstract: A finitely generated group has subexponential growth if the number of group elements expressible as words of length $\le n$ grows subexponentially in $n$. I will show that every countable group that does not contain a subgroup of exponential growth imbeds in a finitely generated group of subexponential growth. This shows that there are no restrictions on being a subgroup of a group of exponential growth, except the obvious ones. This produces in particular the first examples of groups of subexponential growth containing $\mathbb Q$. This also produces groups of subexponential growth and arbitrarily large distortion in uniformly convex Banach (e.g.\ Hilbert) spaces. This is joint work with Anna Erschler. Yves Benoist (Université Paris Sud, France) Title: Random walk on padic flag varieties. Abstract: According to a theorem of Furstenberg, a Zariski dense probability measure on a real semisimple Lie group admits a unique stationary probability measure on the flag variety. In this talk we will see that a Zariski dense probability measure on a padic semisimple Lie group admits finitely many stationary probability measures on the flag variety and we will classify these measures. This is a joint work with JF. Quint. Mladen Bestvina (University of Utah, USA) Title: On the asymptotic dimension of a curve complex. Martin Bridson (University of Oxford, England) Title: Profinite isomorphism problems. Thomas Delzant (Université de Strasbourg, France) Title: Holomorphic families of Riemann surfaces from the point of view of asymptotic group theory. Abstract: We use standard methods of asymptotic group theory (asymptotic cones, limit groups), as well as recent results on the mapping class group to study holomorphic families of Riemann surfaces. Rostislav Grigorchuk (Texas A&M University, USA) Title: Invariant random subgroups of groups of the lamplighter type and some geometric groups. Abstract: After a short introduction to invariant random subgroups (IRS) I will present some results obtained in collaboration with L.Bowen, R.Kravchenko and T.Nagnibeda and with M.Benli and T.Nagnibeda. First I will talk about IRS of groups of lamplighter type. Then about IRS of groups of intermediate growth. And finally, about IRS on hyperbolic groups, mapping class group and group of outer automorphisms of a free group. The latter results are based on study of characteristic random subgroups of a free abelian group of infinite rank and of a free noncommutative group. Joel Friedman (University of British Columbia, Canada) Title: Sheaves on Graphs, L^2 Betti Numbers, and Applications. Abstract: Sheaf theory and (co)homology, in the generality developed by Grothendieck et al., seems to hold great promise for applications in discrete mathematics. We shall describe sheaves on graphs and their applications to (1) solving the Hanna Neumann Conjecture, (2) the girth of graphs, and (3) understanding a generalization of the usual notion of linear independence. It is not a priori clear that sheaf theory should have any bearing on the above applications. A fundamental tool is what we call the "maximum excess" of a sheaf; this can be defined quite simply (as the maximum negative Euler characteristic occurring over all subsheaves of a sheaf), without any (co)homology theory. It is probably fundamental because it is essentially an L^2 Betti number of the sheaf. In particular, Warren Dicks has given much shoter version of application (1) using maximum excess alone, strengthening and simplifying our methods using skew group rings. This talk assumes only basic linear algebra and graph theory. Part of the material is joint with Alice Izsak and Lior Silberman. Gilbert Levitt (University of Caen, France) Title: Vertex finiteness for relatively hyperbolic groups. Abstract: Given a finitely generated group G, we consider all splittings of G over subgroups in a fixed family (such as finite groups, cyclic groups, abelian groups). We discuss whether it is the case that only finitely many vertex groups appear, up to isomorphism. (Joint work with Vincent Guirardel) Elon Lindenstrauss (Hebrew University, Jerusalem) Title: Spectral gap, random walks on Euclidean isometry groups, and selfsimilar measures. Abstract: In contrast to the two dimensional case, in dimension $d \geq 3$ averaging operators on the $d1$sphere using finitely many rotations, i.e. averaging operators of the form $Af(x)= S^{1} \sum_{\theta \in S} f(s x)$ where $S$ is a finite subset of $SO (d)$, can have a spectral gap on $L^2$ of the $d1$sphere. A result of Bourgain and Gamburd shows that this holds, for instance, for any finite set of elements in $\SO (3)$ with algebraic entries and spanning a dense subgroup. We prove a new spectral gap result for averaging operators corresponding to finite subsets of the isometry group of $R^d$, which is a semidirect product of $SO (d)$ and $R^d$, provided the averaging operator corresponding to the rotation part of these elements have a spectral gap. This new spectral gap result has several applications, in particular to the study of self similar measures in $d \geq 3$ dimensions, and can be used to give sharpening to a previous result of Varju regarding random walks on $R^d$ using the elements of the isometry group. Joint work with Peter Varju Alex Lubotzky (Hebrew University of Jerusalem, Israel) Title: Quantum error correcting codes and asymptotic properties of 4dimensional arithmetic hyperbolic manifolds. Abstract: A family of quantum error correcting codes (QECC) is constructed out of congruence quotients of the 4 dimensional hyperbolic space. The parameters of these codes are expressed by some aymptotic properties of the manifolds and their fundamental groups. By estimating these parameters, we disprove a conjecture of Zémor who predicted that such homological QECC do not exist. All notions will be defined and explained. A joint work with Larry Guth. Nicolas Monod (EPFL, Switzerland) Title: Cutting and pasting: a group for Frankenstein. Abstract: I will describe elementary and concrete examples of nonamenable groups without free subgroups. Amos Nevo (Technion, Haifa, Israel) Title: On best possible rates of Diophantine approximation by lattice orbits. Abstract: We consider the orbits of lattice subgroups of semisimple groups acting on homogeneous spaces. We will give general lower and upper bounds on the rate of approximation of a point on the space by a generic orbit. We will then give a sufficient criterion of when these bounds match, and describe many cases in which the criterion holds. This yields best possible results in a host of natural Diophantine approximation problems on homogeneous algebraic varieties. Based on joint Work with A. Ghosh and A. Gorodnik Nikolay Nikolov (University of Oxford, England) Title: Right angled cocompact lattices in higher rank simple Lie groups. Abstract: Let G be a simple Lie group of real rank at least 2 and let L be a lattice in G. When G/L is not compact then L is known to have fixed price 1 and rank gradient zero. The main purpose of this talk is to give examples of cocompact lattices with the same property. This is joint work with M. Abert and T. Gelander. Alexander Olshanskii (Vanderbilt University, USA and Moscow State University, Russia) Title: Relative growth of subgroups in finitely generated groups. Abstract: Let $H$ be a subgroup of a finitely generated group $G$. The (relative) growth function $f(n)$ of $H$ with respect to a finite set $A$ generating $G$, is given by the formula $f(n) = card \{g\in H; g_A \le n\}$. I want to review some recent results on the asymptotic behavior of relative growth functions in free, solvable and other groups. Leonid Potyagailo (Université de Lille 1, France) Title: Similar Relatively hyperbolic actions of a group. Abstract: This is a joint work with Victor Gerasimov (University of Belo Horisonte, Brasil). Let a discrete group $G$ possess two convergence actions by homeomorphisms on compacta $X$ and $Y$. Consider the following question: does there exist a convergence action of $G$ on a compactum $Z$ and continuous equivariant maps $X\leftarrow Z\to Y$? We call the space $Z$ (and action of $G$ on it) the pullback space (action). In such general setting a negative answer follows from a recent result of O.~Baker and T.~Riley. Suppose, in addition, that the initial actions are relatively hyperbolic that is they are nonparabolic and the induced action on the space of distinct pairs of points is cocompact. In the case when $G$ is finitely generated the universal pullback space exists by a theorem of V. Gerasimov. We show that the situation drastically changes already in the case of countable nonfinitely generated groups. We provide an example of two relatively hyperbolic actions of the free group $G$ of countable rank for which the pullback action does not exist. Our main result is that the pullback space exists for two relatively hyperbolic actions of any group $G$ if and only if the maximal parabolic subgroups of one of the actions are dynamically quasiconvex for the other one. We study an analog of the geodesic flow for a large subclass of convergence groups including the relatively hyperbolic ones. The obtained results imply the main result and seem to have an independent interest. Elyahu Rips (Hebrew University of Jerusalem, Israel) Title: Free Engel Groups and Similar Groups. Abstract: This is a joint work with Arye Juhasz. The free nEngel group is defined by the identical relation [x, y,...,y] = 1, y being repeated n times. An example of a "similar" group is the relatively free group with the identical relation [x, y, x, y,...,x, y] = 1. We study the free nEngel group for n >= 40. Our study is based on the generalized small cancellation theory in the version developed in [Rips, 1982, Israel Math. J.]. In order to apply this theory, we need to understand the structure of the contiguity diagrams in the ranked van Kampen diagrams. As our main tool, we introduce a (sophisticated) canonical form of elements. The gradual process of choosing the canonical form leads us to certain combinatorial configurations we call "fully symmetric chains". We use these fully symmetric chains to describe flats in the corresponding Cayley graphs and the contiguity diagrams. Mark Sapir (Vanderbilt University, USA) Title: The Tarski numbers of groups. Abstract: The Tarski number of a nonamenable group is the minimal number of pieces in a paradoxical decomposition of the group. It is known that a group has Tarski number 4 if and only if it contains a free noncyclic subgroup, and the Tarski numbers of torsion groups are at least 6. It was not known whether the set of Tarski numbers is infinite and whether any particular number >4 is the Tarski number of a group. We prove that the set of possible Tarski numbers is infinite even for 2generated groups with property (T), show that 6 is the Tarski number of a group (in fact of any group with large enough first L_2Betti number), and prove several results showing how the Tarski number behaves under extensions of groups. This is a joint work with Mikhail Ershov and Gili Golan. Zlil Sela (Hebrew University of Jerusalem, Israel) Title: Envelopes and equivalence relations in a free group. Abstract: We study and classify all the definable equivalence relations in a free (and a torsionfree hyperbolic) group. To do that we associate a Diophantine set with every definable set, that contains the definable set, and its generic points are contained in the definable set. We then use this Diophantine set to uniformly parametrize the equivalence classes of a definable equivalence relation. Karen Vogtman (Cornell University, USA) Title: Outer space for rightangled Artin groups. Abstract: Rightangled Artin groups interpolate between free groups and free abelian groups, so one may think of their outer automorphism groups as interpolating between Out(F_n) and GL(n,Z). I will describe an Outer space for these automorphism groups which is a hybrid of the Outer space of a free group and the classical symmetric space SL(n)/SO(n). This is joint work with Ruth Charney. John S. Wilson (University of Oxford, England) Title: Metric ultraproducts of finite simple groups Abstract: Metric ultraproducts of structures have arisen in a variety of contexts. The study of the case when the structures are finite groups is recent and motivated partly by the connection with sofic groups. We report on current joint work with Andreas Thom on the topological and algebraic properties of metric ultraproducts of finite simple groups. Partially supported by:

Asymptotic properties of groups
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