Schedule and Abstracts

All talks will take place in Amphithéâtre Hermite, Institut Henri Poincaré.

Talks will be broadcasted: http://omnilive.ihp.fr:8080/player/?m=~5e134b7d65a9611679e03407

Chairpersons

Monday morning: Shaked Bader

Monday afternoon: Alina Vdovina

Tuesday morning: Paige Helms

Tuesday afternoon: Jingyin Huang

Wednesday morning: Macarena Arenas

Wednesday afternoon: Peter Kropholler

Thursday morning: Wolfgang Woess

Thursday afternoon: Alain Valette

Friday morning: Markus Steenbock

Laurent Bartholdi: Automatic actions and equivalence relations

Automatic actions on a Cantor set are at the heart of at least three domains of mathematics: groups of intermediate growth; iteration of a complex rational function, by encoding its Julia set; and substitutional subshifts. I will explain how the orbit structure of such an action may be encoded by automata, and in this manner answer questions from complex and symbolic dynamics, related to decompositions and minimality.

Colin Bleak: Decomposing elements of the one sided shift

We explore properties of the group Aut(X_n^Z,\sigma) of automorphisms of the full one-sided shift on n letters. After giving a bit of history for this well known group, we explain a very fast algorithm to decompose any element of the shift as products of particularly nice finite order elements. This replicates a result of Boyle, Franks, and Kitchens, but our algorithm is (significantly) faster than the previously known one.

Joint with Peter Cameron and Feyisayo Olukoya.

Adrien le Boudec: On the geometry of graphs of actions of solvable groups

Let G be a finitely generated group. To every action of G is associated the Schreier graph of the action. We consider all Schreier graphs of G as a whole, and we are interested in geometric properties common to all of them. In the talk we will be interested in lower bounds for their growth. For large classes of solvable groups, we are able to give explicit lower bounds, that are sharp for many motivating examples. Our approach consists in exhibiting certain subsets L of G, that roughly speaking have the property that every finite configuration of L appears somewhere in the graph of every faithful G-action. The study of such subsets is closely related to the study of confined subgroups of G, which are a natural generalization of uniformly recurrent subgroups (URS) introduced by Glasner-Weiss.

This is joint work with Nicolas Matte Bon.

Jeremie Brieussel: A flexible family of amenable groups

This talk will be devoted to a flexible family of amenable groups that were introduced in a joint work with Tianyi Zheng. The isoperimetric profile, the compression or the speed of random walk can be prescribed to follow a given arbitrary function (sufficiently regular). Variants on the main construction permit to obtain large families of solvable groups, or groups with Shalom's property H_FD. This family of groups was also used by Corentin Le Coz to construct new behaviours of the separation and Poincaré profiles of groups, and by Amandine Escalier to obtain measure equivalence couplings which are quantitatively optimal with respect to isoperimetric constraints.

Yves de Cornulier: On the space of ends of infintely generated groups

The space of ends of a groups was defined by Freudenthal in 1930 for finitely generated groups, and the definition was extended by Specker in 1950 to arbitrary groups. Freudenthal and Hopf proved in the 1940s that each finitely generated group the space of ends is either empty, a singleton, two points, or a Cantor set, and furthermore they characterized 2-ended groups as infinite virtually cyclic groups. Specker then proved that every infinitely generated group has either 1 or infinitely many ends. Groups with infinitely many ends were characterized by Stallings in 1968 in terms of amalgams/HNN extensions, and this was extended to arbitrary groups by Dicks and Dunwoody to all non-locally-finite groups; nevertheless, this does not describe the topology of the space when it is infinite.

This work is concerned with the problem of determining the topology of the space of ends of a group with infintely many ends. We check that it has no isolated point. Unlike the case of finitely generated groups, it is not metrizable in general, e.g., for a free group of infinite rank. We present some topological alternatives for these spaces, discussing separability and cellularity (maximal number of pairwise disjoint open subsets).

Bruno Duchesne: Dendrites and Groups

Dendrites are topological objects that can be thought as real trees of which we forget the distance and retain only the topology. The goal of this talk is to give a panorama about groups acting by homeomorphisms on dendrites and exhibit new topological groups with very interesting properties.

Dominik Francoeur: On some properties of groups generated by bireversible automata

Among self-similar groups, groups generated by bireversible automata are particularly intriguing from the point of view of geometric group theory, since they have, among others, connections with CAT(0) square complexes and commensurators of free groups in the automorphism groups of regular trees. However, currently, they remain quite mysterious, since very few examples of such groups are known. In an effort to get a better grasp on the kind of groups that can or cannot be generated by bireversible automata, we will investigate some of their properties. In particular, we will discuss the distortion of their cyclic subgroups.

Josh Frisch: Harmonic functions on Linear Groups

The Poisson boundary of a group is a probabilistic object which serves a dual role. It represents the space of asymptotic trajectories a random walk might take and it represents the possible space of bounded harmonic functions on a group.

In this talk I will discuss the Poisson boundary for linear groups. In particular I will focus on the question of when the boundary is trivial in which case all bounded harmonic functions on the group are constant. This is joint work with Anna Erschler.

Alejandra Garrido: Piecing together simple topological groups

Piecewise full groups (a.k.a. topological full groups) of Cantor space homeomorphisms are well-known for containing examples of infinite, finitely generated simple groups. In the theory of totally disconnected locally compact groups there is a need for examples that are compactly generated, topologically simple and not discrete. I will report on what one can (and cannot) do with piecewise full groups to obtain such examples. The construction generalises well-known prototypes such as groups of almost automorphisms of trees and commensurators of profinite branch groups.

Based on work with C.D. Reid and D. Robertson.

Rostislav Grigorchuk: Groups, subshifts, graphs, and randomness

In my talk I will discuss an approach for randomness of groups and Schreier graphs. Two results in this direction will be formulated. Then I will describe a group construction based on the use of irreducible Schreier subshifts of finite type.

Finally I will explain how subshifts can be used in group theory and try to explain how to gain a group of intermediate growth from the Thues-Morse sequence. Based on joint results with L.Bowen, R.Kravchenko, M.G.Benli, Y.Vorobets.

Victor Guba: On the density of the Cayley graph of R. Thompson’s group F in standard generators

By the density of a finite graph we mean its average vertex degree. For the Cayley graph of a group G with m generators, it is known that G is amenable iff the supremum of densities of its finite subgraphs has value 2m.

For R. Thompson’s group F, the problem of its amenability is a long-standing open question. There were several attempts to solve it in both directions. For the Cayley graph of F in the standard set of group generators {x_0,x_1} there exists a construction due to Jim Belk and Ken Brown. It was presented at 2004. This is a family of finite subgraphs whose densities approach 3.5. Many unsuccessful attempts to improve this estimate led to conjecture that this construction was optimal. This would imply non-amenabilty of F.

Recently the author got an improvement showing that this conjecture turned out to be false. Namely, there exist finite subgraphs in the Cayley graph of F with density strictly exceeding 3.5. This makes more truthful the assumption that F may be amenable.

Yair Hartman: Tight inclusions

We discuss the notion of "tight inclusion" of dynamical systems which is meant to capture a certain tension between topological dynamics and ergodic theory (especially in the context of "boundary actions"). The level of abstraction we consider is wide enough to include results regarding unitary representations.

Joint work with Mehrdad Kalantar

Vadim Kaimanovich: Singularity of stationary measures

Standard fixed point theorems imply that any Markov chain on a compact state space with weak* continuous transition probabilities has a stationary (invariant) distribution. What can one say about the stationary measures of "compound" chains obtained by taking a convex combination or the product of two Markov operators? If these operators have a common stationary measure, then obviously it is also stationary for the compound operator, but what if the original stationary measures are just equivalent? Would the compound chain admit an equivalent stationary measure as well?

I will answer the latter question in the negative. The corresponding examples use the boundary processes associated with random walks on the modular group PSL(2,Z), and amount to exhibiting two step distributions with the property that their harmonic measures belong to the same measure class, whereas the harmonic measures of their convex combinations or convolution happen to be singular. The involved boundary measures are closely related to the classical constructions of Minkowski and Denjoy.

The talk is based on joint work with Behrang Forghani.

Yash Lodha: Second bounded cohomology, left orderability and a question of Navas

In this talk I will describe some new results around the second bounded cohomology of various finitely generated and finitely presented groups acting on the line.

This leads to the solution (in the negative) of the following question of Navas from his ICM proceedings article: Does the vanishing of second bounded cohomology (with trivial real coefficients) imply indicability for a finitely generated left orderable group?

This is joint work with Fournier-Facio.

Mikhail Lyubich: Spectral Renormalization of the Basilica Group

The Basilica Group is the iterated monodromy group associated with the Basilica map z^2-1. The Schur renormalization transformation describes the relation between the densities of states in various scales. For this group, it happens to be a rational map in two variables. We will describe the dynamical structure of this map that yields the laminar structure for the associated Green current (restricted to the real plane). It implies that the density of states for this group is a Cantor set of positive Lebesgue measure.

It is a joint work with E.Bedford, N.-B. Dang, and R.Grigorchuk.

Nicolás Matte Bon: On actions of certain groups on the real line

Given a group G, we are interested in describing the possible actions of G on the real line. I will describe recent results in this setting that apply to various classes of groups, including finitely generated solvable groups, Thompson’s groups and many relatives. These results have an arboreal component, as we shall see that the understanding actions of such groups on real line are naturally leads to study actions on more general one-dimensional tree-like structures.

The talk is based on past and ongoing joint work with J. Brum, C. Rivas and M. Triestino.

Francesco Matucci: Finite Germ Extensions

We introduce a family of groups of homeomorphisms obtained from groups of piecewise linear homeomorphisms by adding finitely many singularities and we prove results about their simplicity, abelianizations and finiteness properties. This family arose naturally in the process of solving a Kourovka notebook question by Bridson and De la Harpe asking whether there exists a finitely presented group containing the additive group~$\mathbb{Q}$ of rational numbers.

Among the examples we construct, we describe two groups $T\mathcal{A}$ and $V\mathcal{A}$ that are simple, two-generated, finitely presented and contain, respectively, all countable torsion-free abelian groups and all countable abelian groups, explicitly realizing the Boone--Higman embedding theorem. Moreover, we show that they have type~$\Finfty$.

We also discuss how our results can be applied to other related groups, such as some Nekrashevych groups, a class of groups which are generated by Thompson groups V_{n,r} and suitable self-similar groups.

This is joint work with Jim Belk and James Hyde.

Ivan Mitrofanov: Orders on metric spaces and invariants

Let $M$ be a metric space and let $T$ be a total ordering of its points.

For a finite subset $X\subset M$ we calculate the minimal length $l_{opt}(X)$ of a path visiting all its points, the length $l_T(X)$ of the path which visits the points of $X$ with respect to the order $T$, and the ratio $r_T(X) = l_T(X)/l_{opt}(X)$.

As J. Bartholdi and J.Platzman noticed in 1982, for the square $[0;1]^2$ and the order corresponding to the self-similar Peano curve all such ratios $r_T(X)$ are bounded by a logarithmic function of $|X|$.

For a given metric space the existence of such "good" orders is connected with more traditional properties and invariants, such as hyperbolicity, Assouad-Nagata dimension, number of ends and doubling.

The talk is based on joint works with Anna Erschler.

Nicolas Monod: Certain groups with self-similarity properties are acyclic in the bounded sense

We will first recall how cohomological vanishing properties can be a sign that a group is "small"; for instance finite. Analogously, using bounded cohomology one can characterise amenability instead of finiteness.

However, it has been known at least since the 1970s that some very large groups are also "acyclic", that is, have vanishing cohomology with trivial coefficients.

We establish a stronger version of this fact for bounded cohomology that applies to many groups with self-similarity properties, such as Thompson's group F.

Volodymyr Nekrashevych: Dimensions of self-similar groups

We will discuss Hausdorff, conformal, and topological dimensions of the limit space of contracting self-similar groups and related notions, such as contraction coefficients, complexity function, quasi-invariant measures, and growth.

Marialaura Noce: Ramification structures for quotients of the Grigorchuk groups

Groups associated to surfaces isogenous to a higher product of curves can be characterized by a purely group-theoretic condition, which is the existence of the so-called ramification structure. In this talk, we will present recent developments involving ramification structures, such as new links to groups acting on rooted trees.

This is joint work with A. Thillaisundaram.

Mikael de la Salle: Cayley graphs with few automorphisms

The subject of this talk will be automorphisms of Cayley graphs. They always contain the translations by the elements of the group, but the automorphism group can be much larger (think of a the standard Cayley graph of free groups). Given a finitely generated group, we will be interested in how and when a careful choice of a generating set can make the automorphism group very small. In particular, I will explain that every finitely generated group admits a Cayley graph with countable automorphism group.

Based on joint works with Paul-Henry Leemann.

Omer Tamuz: A quantitative Neumann lemma for finitely generated groups

With Elia Gorokhovsky and Nicolás Matte Bon.

We study the coset covering function C(r) of a finitely generated group: the number of cosets of infinite index subgroups needed to cover the ball of radius r. We show that C(r) is of order at least root r for all groups. Moreover, we show that C(r) is linear for a class of amenable groups including virtually nilpotent and polycyclic groups, and that it is exponential for property (T) groups.

Alexander Teplyaev: Spectral analysis on self-similar graphs, fractals, and groups

The talk will present the background and new results on spectral analysis on self-similar graphs and their fractal limits. This topic is related to questions about the spectrum of the Laplacian and random walks associated with some self-similar groups. We will discuss spectral dimensions, self-similar random walks and their diffusion limits, and the role of symmetries and finite ramification in computing the spectrum explicitly. Examples with the pure point or singularly continuous spectrum are of particular interest.

Anitha Thillaisundaram: The Hausdorff dimensions of branch groups

The concept of Hausdorff dimension was defined in the 1930s and was originally applied to fractals and shapes in nature. However, from the work of Abercrombie, Barnea and Shalev in the 1990s, the computation of the Hausdorff dimensions in profinite groups has been made possible. Starting with Abert and Virag's well-known result that there are groups acting on a rooted tree with all possible Hausdorff dimensions, mathematicians have been interested in computing the Hausdorff dimensions of explicit families of groups acting on rooted trees, and in particular, of the so-called branch groups. Branch groups first appeared in the context of the Burnside problem, where they delivered the first explicit examples of finitely generated infinite torsion groups. Since then, branch groups have gone on to play a key role in group theory and beyond. In this talk, we will survey known results concerning the Hausdorff dimensions of branch groups, in particular mentioning some recent joint work Gustavo Fernandez-Alcober and Sukran Gul.

Tiany Zheng: Limit theorems for some long range random walks on nilpotent groups

We consider a natural class of long range random walks on nilpotent groups and develop limit theorems for these walks. The limiting process lives on a nilpotent Lie group which carries an adapted dilation structure and a stable-like process which appears as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the random walk. In addition, the Donsker-type functional limit theorem is complemented by a local limit theorem.

Joint with Z.Q. Chen, T. Kumagai, L. Saloff-Coste and J. Wang.

All talks will take place in Amphithéâtre Hermite, Institut Henri Poincaré.

Talks will be broadcasted on Zoom: https://us02web.zoom.us/j/84067977468?pwd=TzJsR0ZRNVFHQ25md3l6NkJyUjQ5Zz09