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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 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