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# Ergodic and Geometric Group Theory EGG

Seminars — EGG.

More upcoming events on the EPFL memento and elsewhere in Switzerland.
The EGG group meeting is Thursdays at 10:30.

Click title for abstract

• Thursday 27 February 2020, 11:00 at MA B1 524
Wouter Van Limbeek (UIC)
Do thin groups have discrete commensurators?

Let G be a simple Lie group and \Gamma < G a lattice. In 1974, Margulis proved that if the commensurator of \Gamma is dense, then \Gamma is arithmetic. In 2015, Shalom asked if the same is true only assuming \Gamma is Zariski-dense in G. I will report on recent progress on this question, using ideas from infinite ergodic theory, profinite actions, representation theory, Brownian motion, random walks and Hodge theory. I will attempt to give a picture of how all these ideas combine to give information on commensurators. This is joint work with D. Fisher, T. Koberda, and M. Mj.

• Thursday 16 January 2020, 13:00 at MA B1 524
Valeriano Aiello (Geneva)
Knots, subgroups, and representations of the Thompson groups

A few years ago Vaughan Jones introduced a method that yields unitary representations of the Thompson groups, some of which are related to notable graph and knot invariants. In this seminar I will present some results on the Thompson groups as knot constructors, on the subgroups associated with a family of Jones’s representations and a classification of the corresponding spectral measures. This talk is based on joint works with A. Brothier, R. Conti, T. Nagnibeda, V. Jones.

• Thursday 14 November 2019, 13:00 at MA 10
Maxime Gheysens (Dresden)
A Group from Beyond Infinity

The order topology on the first uncountable ordinal yields a fascinating topological space, well known for featuring "pathological" phenomena (such as sequential compactness without compactness, or a Stone–Čech compactification reduced to Alexandrov's one-point compactification). In this talk, we introduce its homeomorphism group, which also comes with its own niceties. For instance, it is an amenable topological group which is strongly allergic to any idea of distance: all its continuous isometric actions have bounded orbits and all its continuous morphisms to metrisable groups are trivial.

• Thursday 31 October 2019, 13:00 at MA 10
Sebastian Hensel (München)
Quasi-morphisms on surface diffeomorphism groups

We will construct nontrivial quasimorphisms on the group of diffeomorphisms of a surface of genus at least 2 which are isotopic to the identity. This involves considering the graph whose vertices correspond to curves on the surface (not up to isotopy!), and transferring usual curve graph methods to this setting. In particular, we show that it is hyperbolic, and we construct elements of Diff0(S) which act as independent enough hyperbolic elements on it. As a consequence, we also solve a question by Burago- Ivanov-Polterovich on the unboundedness of the fragmentation norm. This is joint work with Jonathan Bowden and Richard Webb.

• Wednesday 23 October 2019, 16:00 at MA 30
Gabor Elek (Lancaster)
Free amenable actions of non-amenable groups

The talk is about the various (Measurable, Borel, Continuous) notions of amenability of free amenable actions. My main goal is to sketch the notion of a "canonical" amenable action of non-amenable groups.

• Thursday 4 July 2019, 14:00 at MA 12
David Fisher (Indiana)
Arithmeticity, superrigidity and totally geodesic submanifolds

I will explain a recent result on arithmeticity of hyperbolic manifolds with "many" totally geodesic submanifolds. This is joint work with Bader, Miller and Stover.

• Tuesday 25 June 2019, 15:00 at MA 12
Waltraud Lederle (ETHZ)
Almost automorphisms of trees and completions of Thompson's group V

Thompson's group V was among the first examples of a finitely presented, infinite, simple group. Caprace and De Medts discovered that it can be written as a group of almost automophisms of the binary tree, allowing to embed it densely into a totally disconnected, locally compact (t.d.l.c.) group. It was a question by Le Boudec and Wesolek whether this is the only locally compact group into which it embeds with dense image. Usung Matui's theory of topological full groups associated to one-sided shifts, we find countably many different t.d.l.c. groups containing a dense copy of V. All mysterious words from the abstract will be explained.

• Tuesday 28 May 2019, 16:00 at MA B3 514
Sam Kim (Seoul)
Non-freeness of certain two-parabolic groups

For each positive rational number q, we study the (non-)freeness of the group G(q) generated by 2x2 matrices a = ( (1,0), (1,1) ) and b = ( (1,q), (0,1) ) in SL(2,Q). We give a computational criterion which allows us to prove that if q=s/r for s≤27 then G(q) is non-free, with the possible exception of s=24. In this latter case, we prove that the set of positive integers r for which G(24/r) is non-free has natural density 1. In the course of the proof, it will follow that for a fixed s, there are arbitrarily long sequences of consecutive denominators r such that G_(s/r) is non-free. For the case s > 27, we describe a density estimate. (Joint work with Thomas Koberda)

• Monday 27 May 2019, 14:00 at CM 1 106
Tianyi Zheng (UC San Diego)
Double commutator lemma for invariant random subgroups

We show a generalization of the double commutator lemma for normal subgroups to invariant random subgroups (IRS). This lemma allows a unified approach to classification of IRSs in several examples of groups acting by homeomorphisms on the Cantor set.

• Thursday 23 May 2019, 14:00 at MA 12
László Márton Tóth (EPFL)
Invariant Random Subgroups in groups acting on rooted trees

An invariant random subgroup (IRS) of a countable group is a conjugation invariant probability distribution on the space of subgroups. There have been a number of recent papers studying IRS’s in various types of groups: lattices in Lie groups, the group of finitary permutations of a countable set, free groups, lamplighters and more.
In this talk we will consider IRS’s in groups acting of rooted trees, in particular the group of finitary automorphisms of a d-ary rooted tree. We exploit the action of these groups on the boundary of the tree to understand fixed point sets of ergodic IRS’s. We show that in the fixed point free case IRS’s behave like the ones in lattices in Lie groups, but if there are fixed points they resemble the ones in the finitary permutation group. Joint work with Ferenc Bencs.

• Tuesday 21 May 2019, 16:00 at MA 30
Alex Bearden (Tyler)
A module version of the weak expectation property

An operator space X is said to have the weak expectation property (WEP) if the canonical inclusion of X into its second dual factors through an injective operator space. We introduce a module version of the WEP for operator modules over completely contractive Banach algebras A. We prove many general results (for example, characterizing the A for which A-WEP implies WEP) and also focus particularly on the cases when A=L^1(G) and A=A(G) for a locally compact group G. A highlight of our work is a locally compact analogue of Lance's famous result for discrete groups that amenability is equivalent to WEP for the reduced C*-algebra. This is joint work with Jason Crann and Mehrdad Kalantar.

• Thursday 16 May 2019, 13:00 at GR A3 30
Quasi-regular representations of discrete groups and associated C*-algebras

We introduce several equivalence relations on the set of subgroups of a countable group G, defined in terms of the quasi-regular representations, and present some rigidity results in terms of these equivalence relations for certain classes of subgroups. Furthermore, we present some results concerning the ideal structure of the C*-algebras generated by the quasi-regular representations. This is joint work with Bachir Bekka.

• Thursday 28 March 2019, 13:00 at GR A3 30
Galois action on compact hyperbolic surfaces and their associated solenoids

Let S be a compact hyperbolic surface uniformised by a Fuchsian group Γ. For any element σ ∈ G=Gal(C/Q) the natural Galois action of G on the coefficients of the algebraic equation corresponding to the Riemann surface S yields a new hyperbolic surface Sσ with uniformising group Γσ.
Little seems to be known about the relationship between Γ and Γσ as subgroups of PSL_2(R). I will attempt to show that by studying the action of G on the solenoid associated to S one can find some invariants of this Galois action. I will apply these results to present explicit (arithmetic) Fuchsian groups that uniformise non-Galois-conjugate hyperbolic surfaces.

• Thursday 21 March 2019, 13:30 at GR A3 30
Bruno Duchesne (Nancy)
Maximal representations of complex hyperbolic lattices in infinite dimension.

Unlike lattices in higher rank, lattices of simple Lie groups in rank 1 are not rigid. This gives rise to the Teichmüller spaces for example.
For representations of lattices of the isometry group of the complex hyperbolic lattices in Hermitian Lie groups, the Kähler form yields a numerical invariant, the Toledo number. When this number is maximal, these representations are rigid when the dimension is at least 2.
We will focus on infinite dimensional representations of complex hyperbolic lattices that are not unitary but preserve a Hermitian form of finite index. This gives actions by isometries on infinite dimensional Hermitian symmetric spaces and one can define a Toledo number as well.
We will see that for surface groups, one can create maximal representations that do not preserve any finite dimensional subspace. Conversely, for complex hyperbolic lattices in dimension at least 2, these maximal representations factor through a finite dimensional Lie subgroup.

• Thursday 7 March 2019, 13:00 at GR A3 30
Andy Zucker (Paris)
Maximally highly proximal flows

Given a Polish group G, the maximally highly proximal flows, or MHP flows, are those flows which do not admit a non-trivial highly proximal extension. Examples of MHP flows include the Samuel compactification, the universal minimal flow, the universal minimal proximal flow, and the Furstenberg boundary. In this talk, we will seek a detailed understanding of topologically transitive MHP flows. In particular, if X is an MHP flow with a comeager orbit, then X is the Samuel compactification of G/H for some closed subgroup H, and in many cases, we can describe H explicitly.

• Thursday 15 November 2018, 15:00 at MA 10
Gabor Elek (Lancaster)
Qualitative graph limits and the space of Cantor actions

Recently, I defined a notion of limit for finite Schreier graphs. It turns out that the natural limit objects are stable continuous group actions on the Cantor set. The limit notion leads to the definition of qualitative weak equivalence (this notion has already been defined by Matui and Li for minimal actions of the integers). I will show some results on the space of weak equivalence classes and will allude to some applications in the theory of local distributed algorithms.

• Thursday 1 November 2018, 15:00 at MA 10
Rachel Skipper (Göttingen)
Finiteness properties of simple groups

A group is said to be of type F_n if it admits a classifying space with compact n-skeleton. We will consider the class of Röver–Nekrachevych groups, a class of groups built out of self-similar groups and Higman–Thompson groups, and use them to produce a simple group of type F_{n-1} but not F_n for each n. These are the first known examples for n≥3. As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups, the first is due to Caprace and Rémy.
This is a joint work with Stefan Witzel and Matthew C. B. Zaremsky.

• Wednesday 31 October 2018, 15:00 at MA 30
Maria Gerasimova (Dresden)
Unitarisability of discrete groups

A group G is called unitarisable if every uniformly bounded representation of G in a Hilbert space can be conjugated to a unitary representation. It is well known that amenable groups are unitarisable. It has been open ever since whether this characterises the unitarisability of a group.
One of the approaches to study unitarisability is related to the space of Littlewood functions T1(G). We define the Littlewood exponent Lit(G) of a group G as follows:
Lit(G) = inf { p : T1(G) ⊆ lp(G) }.
On the one hand, Lit(G) is related to unitarisability and amenability and, on the other hand, it is related to some geometry of G or, more precisely, to the behavior of Cayley graphs when one increases the generating sets of G. We will discuss some applications of this connection.
We will also discuss the notion of p-isometrisability for uniformly bounded representations on p-spaces, which coincides with the usual unitarisability if p = 2. We will discuss examples of p-isometrisable groups and mention open questions and conjectures.
This is joint work with Dominik Gruber, Nicolas Monod and Andreas Thom.

• Wednesday 24 October 2018, 15:00 at MA 30
Carlos De la Cruz Mengual (ETHZ)
Stability in bounded cohomology for classical groups

Homological stability is a phenomenon that has been studied and established in the context of ordinary group homology for several infinite ascending series of groups. So far the only existing stability result known for bounded cohomology, by Monod, concerned the families of the general and special linear groups over any local field. In this talk we present an argument that proves stability for the symplectic families over the fields of real and of complex numbers. We first describe a general method that guarantees bounded-cohomological stability along a series of locally compact second-countable groups, provided that there exists a family of highly connected complexes on which the groups have a highly transitive action. Then, we introduce a new family of complexes associated to the symplectic groups, which we call symplectic Stiefel complexes. Similar kinds of objects can be defined for other families of classical groups.
This is joint work with Tobias Hartnick.

• Wednesday 17 October 2018, 15:00 at MA 30
François Fillastre (CNRS)
Hyperbolic geometry of shapes of convex bodies

We define a distance on the space of convex bodies in then-dimensional Euclidean space, up to translations and homotheties, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient Lorentzian structure is an extension of the intrinsic area form of convex bodies. We deduce that the space of shapes of convex bodies (i.e. convex bodies up to similarities) has a proper distance with curvature bounded from below by −1. In dimension 3, this space naturally identifies with the space of distances with non-negative curvature on the 2-sphere.
(joint work with C. Debin)

• Thursday 4 October 2018, 15:00 at MA 10
James Hyde (St. Andrews)
The group of asynchronus rational automata is simple but not finitely generated

The group R of asynchronus rational automata was introduced by Grigorchuk, Nekrashevych, and Sushchanskii in 2000. Jim Belk, Francesco Matucci and myself show in a joint paper that R is simple but not finitely generated. I will define R and sketch the proofs of these two results.

• Monday 4 June 2018, 16:30 at MA 31
Yusuke Isono (RIMS, Kyoto)
Introduction to amalgamated free product von Neumann algebras

The aim of this talk is to give a brief introduction of amalgamated free products (AFP) in von Neumann algebra theory. I will first talk about definitions, basic properties, and some concrete examples. I then focus on the factoriality problem for AFP and explain my recent work (joint with C. Houdayer) on it.​

• Thursday 31 May 2018, 13:00 at MA 30
Thomas Haettel (Montpellier)
Hyperbolic rigidity of higher rank lattices

We will show that every action by isometries of a higher rank lattice on a Gromov-hyperbolic space is elementary. Among consequences, we obtain another proof of the Farb--Kaimanovich--Masur Theorem that any morphism from a higher rank lattice to a mapping class group has finite image. Guirardel and Horbez also deduce another proof of the Bridson--Wade Theorem that any morphism from a higher rank lattice to Out(Fn) has finite image.​

• Thursday 24 May 2018, 13:00 at MA 30
Cyril Houdayer (Orsay)
Strong ergodicity versus spectral gap for group actions on measure spaces

It is well-known that for probability measure preserving (pmp) group actions, if the associated Koopman representation has spectral gap (i.e. has no almost invariant vectors), then the action is strongly ergodic (i.e. has no non-trivial almost invariant measurable subsets). The converse is however not true as demonstrated by Schmidt’s example.

In this talk, I will present a characterization of strong ergodicity for arbitrary nonsingular group actions in terms of a spectral gap property of the full group of the associated orbit equivalence relation. I will then explain how this criterion can be used to characterize strong ergodicity of the Maharam extension of group actions of type III in terms of a new invariant, analogous to Connes tau invariant for von Neumann type III factors. This is joint work with A. Marrakchi and P. Verraedt.

• Thursday 17 May 2018, 13:00 at MA 30
Giles Gardam (Technion)
Exotic one-relator groups

Groups admitting a presentation with only one relator enjoy various algebraic and topological properties, and the class includes examples such as surface groups, 2-bridge knot groups, and many free-by-cyclic groups. On the other hand, there are some well-known pathologies such as the Baumslag--Solitar groups and their variations. In this talk I will introduce a family of one-relator groups that have exotic properties despite the absence of such Baumslag--Solitar subgroup pathologies. This family gives counterexamples to questions on when one-relator groups are CAT(0), automatic, cubulated, or residually finite. Joint work with Daniel Woodhouse (arXiv:1711.08755).

• Monday 14 May 2018, 14:00 at MA 12
Christophe Garban (Lyon)
Inverted orbits of exclusion processes, diffuse-extensive-amenability and (non-?)amenability of the interval exchanges

After a brief introduction on the notion of amenability for groups, I will focus on the group of interval exchanges (the IET group) which is believed to be amenable. One of the (many) equivalent criteria to show that a group is amenable is Kesten's criterion on the return probabilities of random walks. In the case of G=IET, a recent work by Juschenko, Matte Bon, Monod and De La Salle provides a new criterion which is also of probabilistic nature. This new criterion involves the size of the inverted orbit of a certain random walk on the wobbling group W(Z^d) of permutations of Z^d. The aim of this talk will be to introduce natural models of random walks on permutations of Z^d for which this criterion can be analyzed.

The talk will not require any prerequisites.

• Wednesday 8 May 2018, 15:00 at MA 31
Masato Mimura (EPFL & Tohoku)
Amenability versus non-exactness of dense subgroups of a compact group

We provide an example of a compact group K, a 5-generated dense subgroup Lambda_1 in K and an element u in K such that - Lambda_1 is amenable, but - the group Lambda_2, generated by Lambda_1 and u, is non-exact, in other words, it fails to have property A of Guoliang Yu. Our construction is built upon a construction of non-exact groups that are LEF by Osajda and Arzhantseva--Osajda.

• Thursday 3 May 2018, 13:00 at MA 30
Yash Lodha (EPFL)
Coherent group actions on the real line or an interval

The subgroup structure of Thompson's group F is quite mysterious and several recent papers (for instance those of Golan, Sapir and Brin, Bleak, Moore) have been devoted to developing a systematic structure theory of subgroups of F. One related prominent result is due to Vaughan Jones, who showed that a specific subgroup of F encodes all oriented knots in a natural manner.
In this talk I will introduce a class of actions on the real line or an interval, which provides a systematic framework for proving non embeddability results for Thompson's group F and generalisations. The class of actions (and the underlying class of groups which admit such actions) is interesting even for its own sake, and one can draw certain algebraic conclusions, and some concerning rigidity of the action, from a fairly simple set of dynamical axioms.

• Thursday 26 April 2018, 13:00 at MA 30
Steffen Kionke (Karlsruhe)
Groups acting on rooted trees and representations on the boundary

Every action of a group on a rooted tree induces an action on the boundary of the tree. This action yields various representations of the group on spaces of functions on the boundary. In this talk we discuss the structure of such representations. After reviewing of the basic concepts, we will introduce locally 2-transitive actions on trees and give examples of such actions. It will be discussed how this property gives rise to an explicit decomposition of the boundary representation into irreducible constituents.

• Thursday 12 April 2018, 13:00 at MA 30
Peter Feller (ETHZ)
Lines in complex groups and knot theory

We introduce uniqueness questions about polynomial embeddings of the complex line C into complex spaces. We will discuss open problems and present a result concerning embeddings into affine algebraic groups. We will take a low-dimensional topology point of view and describe how knots — one-manifolds in 3-manifolds — arise as objects to be considered.
Based on joint work with Immanuel van Santen. No knowledge of knot theory or algebraic geometry assumed. ​

• Thursday 5 April 2018, 13:00 at MA 30
Damian Osajda (Wroclaw)
Residually finite non-exact groups

I will describe a construction of finitely generated residually finite groups that are non-exact (=without Guoliang Yu's property A).
The main tool is the graphical small cancellation technique of Gromov. The construction is based on my earlier construction of groups containing isometrically expanders and on the work of D. Wise on cubulations of small cancellation groups.​

• Thursday 29 March 2018, 16:15 at Bernoulli
Justin Tatch Moore (Cornell)
Subgroups of Thompson's group

Matthew Brin and Mark Sapir have conjectured that every subgroup of Thompson's group $F$ is either elementary amenable or else contains an isomorphic copy of $F$. We hope to address this question by attempting to classify - or at least better understand - the class of all finitely generated subgroups of $F$ up to biembeddability. In particular, it seems necessary to obtain a more complete understanding of the elementary amenable subgroups of $F$ before attempting to prove the Brin-Sapir conjecture. We will show that there is a family of finitely generated elementary amenable subgroups of $F$ which is strictly well ordered in type $\epsilon_0$ by the embeddability relation. Moreover, we show that the EA-class of a finitely generated subgroup of $F$ can be made arbitrarily large below $\epsilon_0$, improving on previous results of Brin. The work presented in this talk is joint with Collin Bleak and Matthew Brin.​

• Thursday 22 March 2018, 17:15 at MA 30
David Kyed (University of Southern Denmark)
Measure equivalence for locally compact groups

In the talk I will discuss the notion of measure equivalence within the framework of locally compact second countable groups, with primary emphasis on the unimodular case, and explain how classical results, such as the Ornstein–Weiss theorem, generalize to this setting. If time permits, I will also discuss the notion of uniform measure equivalence and its relation to coarse equivalence. The talk is based on joint work with Juhani Koivisto and Sven Raum.

• Thursday 8 March 2018, 13:00 at MA 30
Thomas Koberda (Virginia)
Diffeomorphism groups of intermediate regularity

Let $M$ be the interval or the circle. For each real number $\alpha \in [1,\infty)$, write $\alpha=k+\tau$, where $k$ is the floor function of $\alpha$. I will discuss a construction of a finitely generated group of diffeomorphisms of $M$ which are $C^k$ and whose $k^{th}$ derivatives are $\tau$--H\"older continuous, but which are admit no algebraic smoothing to any higher H\"older continuity exponent. In particular, such a group cannot be realized as a group of $C^{k+1}$ diffeomorphisms of $M$. I will discuss the construction of countable simple groups with the same property, and give some applications to continuous groups of diffeomorphisms. This is joint work with Sang-hyun Kim.

• Postponed date, 13:00 at MA 30
Yash Lodha (EPFL)
Coherent group actions on the real line or an interval

The subgroup structure of Thompson's group F is quite mysterious and several recent papers (for instance those of Golan, Sapir and Brin, Bleak, Moore) have been devoted to developing a systematic structure theory of subgroups of F. One related prominent result is due to Vaughan Jones, who showed that a specific subgroup of F encodes all oriented knots in a natural manner.
In this talk I will introduce a class of actions on the real line or an interval, which provides a systematic framework for proving non embeddability results for Thompson's group F and generalisations. The class of actions (and the underlying class of groups which admit such actions) is interesting even for its own sake, and one can draw certain algebraic conclusions, and some concerning rigidity of the action, from a fairly simple set of dynamical axioms.

• Monday 19 February 2018, 13:00 at MA 12
Kei Funano (Tohoku)
Concentration of eigenfunctions of the Laplacian on a closed Riemannian manifold

In this talk I will discuss concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. I will show that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. I will also explain restricted exponential concentration inequalities and Sogge type reverse Hölder inequalities for eigenfunctions. This talk is based on the work with Yohei Sakurai.

• Thursday 21 December 2017, 13:00 at MA 31
Masato Mimura (EPFL and Tohoku)
The space of marked groups and analytic properties

The space of k-marked groups was introduced by Grigorchuk. It is the space of all k-generated groups with marking (the ordered k-generators); it is endowed with a compact metrizable topology (the Cayley topology, in other words, the Chabauty topology with respect to F_k). We will discuss some behavior of analytic properties in terms of this topology, partly in connection to coarse geometry of the (infinite) disjoint union of finite marked groups.

• Thursday 14 December 2017, 13:00 at MA 31
Dawid Kielak (Bielefeld)
Amenability, matrices over group rings, and applications

We will see how amenability of a group assists one when working with its group ring, and also helps in working with group rings in general. We will introduce a notion of the Newton polytope of a matrix, and see how it is relevant in the study of fibreings of 3-manifolds, and in general in the theory of Bieri--Neumann--Strebel invariants.

• Monday December 2017, 13:00 at MA 31
Maxime Gheysens (Dresden)
Propriétés de point fixe affines

Les actions affines permettent de mieux comprendre certains groupes et de capturer des comportements divers (en général de nature analytique ou topologique). En particulier, on peut se demander sous quelles conditions (sur le groupe, sur l'espace ou sur l'action) il est toujours possible de trouver un point fixe. Dans cet exposé, nous passerons en revue quelques propriétés de point fixe en lien avec la finitude, la compacité et la moyennabilité du groupe.

• Thursday 30 November 2017, 13:00 at MA 31
Alessandro Sisto (ETHZ)
Bounded cohomology of acylindrically hyperbolic groups

Acylindrically hyperbolic groups form a vast collection of groups including non-elementary hyperbolic and relatively hyperbolic groups, mapping class groups, Out(F_n), many groups acting on CAT(0) cube complexes, and many others. I will present a few results that describe their bounded cohomology in terms of their so-called hyperbolically embedded subgroups and, time permitting, discuss an expected connection with hyperbolic 3-manifolds.

• Thursday 23 November 2017, 13:00 at MA 31
Alexander Kolpakov (Neuchatel)
Counting subgroups and cellular complexes

We derive a generating series for the number of free subgroups of finite index in certain free products of cyclic groups. We also give a generating series for the number of their conjugacy classes. This computation turns out the same as finding the number of orientable cellular complexes and their isomorphism classes in dimensions two and three. Our solution rests on the connection between a subgroup H of a chosen group G and the action of G on the set of co-sets G/H understood in suitable geometric terms. We provide non-linear recurrence relations for the number of objects of both kinds, and prove that in some cases the generating series is not holonomic, and in all cases it is not algebraic. The talk will focus on the interplay between algebra, geometry and combinatorics related to the above problems. This is joint work with Laura Ciobanu (Heriot-Watt University, UK).

• Thursday 9 November 2017, 13:00 at MA 31
Nicolas Matte Bon (ETHZ)
Actions and homomorphisms of topological full groups

To any group or pseudogroup of homeomorphisms of the Cantor set one can associate a larger (countable) group, called the topological full group. It is a complete invariant of the groupoid of germs of the underlying action (every isomorphism between full groups is implemented by a conjugacy of the pseudogroup). First I'll state a result relating the growth of the orbits of a pseudogroup to a combinatorial fixed point property of its full group, and explain an application related to co-amenability and growth of Schreier graphs of finitely generated groups. Next I will discuss a theorem on the possible actions on topological full groups on compact spaces, and apply it to show that not only isomorphisms but arbitrary homomorphisms between full groups are often implemented at the level of the groupoids. These results are proven working in the Chabauty space.

• Thursday 12 October 2017, 13:00 at MA 31
Jonas Wahl (KU Leuven)
Bernoulli actions of type III and L2-cohomology

While the study of measure preserving Bernoulli actions of discrete groups has yielded many stunning results during recent years, not so much is known as soon as one steps away from the measure preserving case. In fact, until recently, the only discrete group known to admit a Bernoulli action without invariant measure was the group of integers. In this talk, I aim to demonstrate that the question whether or not a given group admits a Bernoulli action without invariant measure, depends strongly on its first L2-cohomology. In particular, I will show how to construct such actions for a large class of discrete groups and give some applications. This is joint work with Stefaan Vaes.

• Friday 6 October 2017, 13:00 at MA 31
Michele Triestino (Dijon)
Ping-pong and orders on free groups

Actions on one-dimensional manifolds are intimately related to invariant orders on groups. In a recent work with Malicet, Mann and Rivas, we study circular orders on free groups through ping-pong actions on the circle. We prove that a free group admits isolated invariant orders if and only if the rank is even. Similarly, F_n X Z admits isolated linear orders if and only if n is even.​

• Thursday 28 Septembre 2017, 13:00 at MA 31
Anastasia Khukhro (Neuchâtel)
Geometry of finite quotients of groups

Geometric properties of a collection of finite quotients of a group can provide information about the group if the set of finite quotients is sufficiently rich. Using a metric space constructed using Cayley graphs of these finite quotients, one can exploit the connections between the world of group theory and graph theory to give examples of metric spaces with interesting and often surprising properties. In this talk, we will describe some results in this direction, and then give recent results concerning geometric rigidity of finite quotients​ of a group (joint work with Thiebout Delabie).​

• Thursday 15 June 2017, 13:00 at MA 12
László Márton Tóth (Budapest)
Uniform rank gradient, cost and local-global convergence (joint with Miklós Abért)

The notion of combinatorial cost for sequences of graphs was introduced by Elek as an analogue of the cost of measure preserving equivalence relations. We show that if a graph sequence is local-global convergent, then its combinatorial cost equals the cost of the limit graphing.
This in particular implies previous results of Elek on combinatorial cost, and gives an alternate proof of the Abert-Nikolov theroem that connects the rank gradient of a chain of subgroups to the cost of its profinite completion.
It also turns out that local-global convergence is a useful tool in reinforcing previous results on the rank gradient. We obtain a uniform continuity result for the rank gradient for Farber sequences of subgroups in groups with fixed price, and show vanishing of the rank gradient in finitely presented amenable groups for arbitrary sequences (with index tending to infinity).

• Monday 8 May 2017, 13:00 at MA 30
Liviu Paunescu (Bucharest)
The Birkhoff—von Neumann Theorem in type II_1 setting

The classic Birkhoff—von Neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices. In this talk, we study a generalisation of this theorem in the type II_1 setting. Namely, we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms, of the unit interval, with similar properties. We show that a weaker version of this theorem still holds.
Joint work with Florin Radulescu.

• Thursday 27 April 2017, 13:00 at MA 12
Sang-Hyun Kim (Seoul)
Obstruction for a virtual C^2 action on the circle

When does a group virtually admit a faithful C^2 action on the circle? We provide an obstruction using a RAAG. Examples include all (non-virtually-free) mapping class groups, Out(Fn) and Torelli groups. This answers a question by Farb. (Joint work with Hyungryul Baik and Thomas Koberda)​

• Thursday 13 April 2017, 13:00 at MA 12
Michele Triestino (Dijon)
What is a lattice in Diff(S1) ?

No doubts that lattices in Lie groups are very interesting objects, one of the reasons being that they naturally carry interesting geometry (the quotient manifold). The groups of diffeomorphisms Diff(M) are also very fascinating, on one side for the many connections to dynamics, on the other side because they are nearly as nice as a Lie group. In fact, they can be considered Lie group, but infinite-dimensional. Among them, the group of circle diffeomorphism Diff(S1) is my favourite (also because for higher dimensional manifolds Diff(M) is a much wilder object). One (phylosophical) problem is to try to give any reasonable sense to the concept of "lattice" in Diff(S1), the main bad issue being that there is no natural notion of measure on it (even if part of my Ph.D. thesis was in this direction). If we agree with the idea that lattices, or simpy discrete subgroups, should give geometry, then we have candidates. This talk will be an excursion through the study of (locally) discrete groups of circle diffeomorphisms, from dynamics to geometry: we will start from the two main classes of examples, Fuchsian groups and Thompson's group, then present a series of partial results, after which we can conjecture that all discrete groups should fit in the two known classes, which is actually one only, if we think of Thompson's group as a piecewise-Fuchsian group. This is based upon a running project with S. Alvarez, P. Barrientos, B. Deroin, D. Filimonov, V. Kleptsyn, D. Malicet, C. Menino and A. Navas.

• Thursday 6 April 2017, 14:00 at MA 12
David Kyed (Odense)
L^2-Betti numbers of universal quantum group

I will report on joint works with Julien Bichon, Sven Raum, Matthias Valvekens and Stefaan Vaes, revolving around the computation of L^2-Betti numbers for universal quantum groups. Among our main results is the fact that the first L^2-Betti number of the duals of the free unitary quantum groups equals 1, and that all other L^2-Betti numbers vanish. All objects mentioned in the abstract will be defined, more or less rigorously, during the talk.

• Thursday 30 March 2017, 14:00 at MA 12
Selçuk Barlak (Odense)
Cartan subalgebras and the UCT problem

The question whether every separable, nuclear C*-algebra satisfies Rosenberg-Schochet's universal coefficient theorem (UCT) is a major open problem in C*-algebra theory. Currently, renewed interest in this so-called UCT problem arises from the recent breakthrough results in the classification program for separable, simple, nuclear C*-algebras, where the UCT plays a rather mysterious role. In this talk, connections between Cartan subalgebras, that is, MASAs admitting faithful conditional expectations and generating the ambient C*-algebras in a suitable sense, and the UCT problem will be illustrated. Using remarkable results of Renault and Tu, we will see that separable, nuclear C*-algebras with Cartan subalgebras satisfy the UCT. Moreover, I will try to explain the close connection between the UCT problem on the one hand and Cartan subalgebras and finite order automorphisms of the Cuntz algebra O_2 on the other. This is joint work with Xin Li.

• Thursday 9 March 2017, 13:00 at MA 12
Todor Tsankov (Paris)
On metrizable universal minimal flows

To every topological group, one can associate a unique universal minimal flow (UMF): a flow that maps onto every minimal flow of the group. For some groups (for example, the locally compact ones), this flow is not metrizable and does not admit a concrete description. However, for many "large" Polish groups, the UMF is metrizable, can be computed, and carries interesting combinatorial information. The talk will concentrate on some new results that give a characterization of metrizable UMFs of Polish groups. It is based on two papers, one joint with I. Ben Yaacov and J. Melleray, and the other with J. Melleray and L. Nguyen Van Thé.

• Wednesday 8 March 2017, 13:00 at MA 30
Yash Lodha (EPFL)
A complex of clusters

In joint work with Justin Moore, we introduced a finitely presentable group G which is nonamenable and does not contain nonabelian free subgroups. To prove that this group is of type $F_{\infty}$, I introduced a CW complex X and an an action of G on X by cell permuting homeomorphisms. The complex X is a topological generalisation of nonpositively curved cube complexes. I will describe this generalisation both for its own sake, as well as a tool to study the group G​.

• Thursday 8 December 2016, 13:00 at MA 31
Boundary 2-transitive automorphism groups of trees

Let T be a locally finite tree all of whose vertices have valency at least 6. In this talk, we will be interested in closed subgroups of Aut(T) acting 2-transitively on the set of ends of T and whose local action at each vertex contains the alternating group. The main result I will present is a full classification of these groups, up to isomorphism. The outcome of the classification is a countable family of group, most of which are new, and all containing a simple subgroup of index ≤ 8. I will also explain what makes the assumption of 2-transitivity on the boundary natural and discuss some direct consequences of the result.

• Thursday 1 December 2016, 15:00 at MA 10
Hyungryul Baik (Bonn)
Laminar groups in geometric topology

Groups acting on the circle with invariant laminations are called laminar groups. The theory of laminar groups was initiated by Danny Calegari who was motivated by the work of Bill Thurston on universal circles for taut foliations. In this talk, we introduce the notion of laminar groups and show that such groups arise very naturally in various contexts in geometric topology.

• Thursday 1 December 2016, 13:00 at MA 31
Sven Raum (EPFL)
On the type I conjecture for groups acting on trees

A locally compact group is of type I -- roughly speaking -- if all its unitary representations can be uniquely written as a direct integral of irreducible representations. This property is of utter importance in the study of Lie groups and algebraic groups. The type I conjecture predicts that every closed subgroup of the automorphism group of a locally finite tree that acts transitively on the boundary of the tree is of type I. A proof of this conjecture would give a new perspective on the representation theory of rank one algebraic groups over non-Archimedean fields and prove a huge class of groups whose representation theory is well-behaved.
I will describe my recent effort to attack the type I conjecture and understand the class of type I groups acting on trees by operator algebraic means.

• Thursday 24 November 2016, 13:00 at MA 31
Stefan Witzel (Bielefeld)
A framework for Thompson groups

Richard Thompson introduced three groups, denoted F, T, V, which are interesting for various reasons. For example, T an V were the first known examples of finitely presented infinite simple groups. Better than being finitely presented they are in fact of type F_\infty. In subsequent years, various groups have been constructed with similar features. They are now jointly referred to as Thompson groups''. Many Thompson groups have been shown to be F_\infty as well and the strategy is always similar. I will present a description of Thompson groups are fundamental groups of Ore categories. Using this framework I will describe how F_\infty-proofs can be treated uniformly up to the point were some insight is needed.​

• Thursday 3 November 2016, 13:00 at MA 31
Corina Ciobotaru (Fribourg)
Finsler compactification of vector spaces

A real vector space V of dimension n admits various compactifications depending on the metric that is considered on V. For example, when V is endowed with the usual Euclidean metric, the corresponding compactification is V union with the n-1 dimensional sphere. In a recent joint work with Linus Kramer and Petra Schwer we study the case of a (not necessarily symmetric) Finsler metric d_F on V. By employing elementary results from model theory and ultraproducts of metric spaces we give an easy proof that the corresponding ''compactification'' of (V, d_F) is V union with the boundary of the dual polyhedron associated with d_F.

• Thursday 20 October 2016, 13:00 at MA 31
Nicolás Matte Bon (ETHZ)
Subgroup dynamics, extreme boundaries, and C*-simplicity

Let G be a countable group. The space of subgroups of G, endowed with the Chabauty topology, is naturally a compact space on which G acts continuously by conjugation.
The talk will focus on the study of the closed, minimal invariant subsets of this action, called uniformly recurrent subgroups (URS's). I will explain a general method to study URS's in a class of groups, that we call "infinitesimally supported groups of homeomorphisms". Examples of groups in this class are Thompson's groups, groups of piecewise projective homeomorphisms, branch groups, groups acting on trees with prescribed action almost everywhere, topological full groups. Applications will be given to C*-simplicity criteria for groups in this class, and to some rigidity properties of their minimal actions on compact spaces.
This is joint work with Adrien Le Boudec.

• Thursday 29 September 2016, 13:00 at MA 31
Yash Lodha (EPFL)
Chain groups of homeomorphisms of the interval and the circle

We introduce the notion of chain groups of homeomorphisms of a one-manifold, which are groups finitely generated by homeomorphisms each sup- ported on exactly one interval in a chain, subject to a certain mild dynamical condition. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator sub- group or the action of the group has a wandering interval. In the latter case, the chain group admits a canonical quotient which is also a chain group, and which has a simple commutator subgroup. Moreover, any 2-chain group is isomorphic to Thompson’s group F. On the other hand, every finitely generated subgroup of Homeo+([0,1]) can be realized as a subgroup of a chain group. As a corollary, we show that there are uncountably many isomorphism types of chain group, as well as uncountably many isomorphism types of countable simple subgroups of Homeo+([0,1]). We also study chain groups of various regularities, and show that there are uncountably many isomorphism types of chain groups which cannot be realized by C2 diffeomorphisms.
This is joint work with Sang-hyun Kim and Thomas Koberda.

• Thursday 15 September 2016, 13:00 at MA 12
Masato Mimura (EPFL)
Superintrinsic synthesis in fixed point properties

For a class X of metric spaces, we say a finitely generated group G has the fixed point property (F_X), relative to X, if all isometric G-actions on every member of X have global fixed points. Fix a class X of "non-positively curved spaces" (for instance, in the sense of Busemann) stable under certain operation. We obtain new criteria to "synthesize" the "partial" (F_X) (more precisely, with respect to subgroups) into the "whole" (F_X). A basic example of such X is the class of all Hilbert spaces, and then (F_X) is equivalent to the celebrated property (T) of Kazhdan.
Our "synthesis" is intrinsic, in the sense of that our criteria do not depend on the choices of X. The point here is that, nevertheless, we exclude all of "Bounded Generation" axioms, which were the clue in previous works by Y. Shalom. As applications, we present a simpler proof of (T) for elementary groups over noncommutative rings (Ershov--Jaikin, Invent. Math., 2010). Moreover, our approach enables us to extend that to one in general L_p space settings for all finite p>1.

### Contacts

Prof. Nicolas Monod