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Abstracts of the lectures

Emmanuel Breuillard. Approximate groups (3 hours)

The notion of approximate group was introduced by Terence Tao a few years ago in order to describe finite subsets A of an ambient group whose product set AA is not much larger than A. A large literature exists on the problem in abelian groups, and only recently did people start to investigate thoroughly the non-commutative case. A major question, the "non-commutative Freiman problem", consists in describing the rough structure of such sets. In this mini-course, I will discuss approximate groups, give the basic tool-kit required to manipulate these objects, and describe works of Helfgott, Hrushovski, Pyber-Szabo and Breuillard-Green-Tao regarding approximate subgroups of GL(n).

Pierre-Emmanuel Caprace. Rank, lattices, and asymptotic cones in nonpositive curvature (3 hours)

The theory of semisimple Lie groups and symmetric spaces highlights a remarkable dichotomy ‘rank one vs. higher rank’, notably illustrated by a large number of rigidity results whose validity typically requires the rank to be greater than one. The goal of these lectures is to illustrate various aspects of this dichotomy in the realm of metric spaces of non-positive curvature and their lattices, beyond the classical case of symmetric spaces. Central to our discussion will be the Rank Rigidity of CAT(0) cube complexes (joint work with Michah Sageev).

Alain Valette. On a-T-menability and its permanence properties (2 hours)

A locally compact group has the Haagerup property, or is a-T-menable, if it admits a metrically proper, isometric action on a Hilbert space. In the first talk, we will indicate why this property is interesting and useful, and how geometry can be used to obtain lots of examples. In the second talk, we will present the following permanence result for a-T-menability: let H, G, X be countable groups, with X a quotient group of G; the permutational wreath product HXG is a-T-menable if and only if each of H, G, and X is a-T-menable. (The if part is due to Cornulier, Stalder and myself and the only if part is due to Chifan and Ioana.)

Abstracts of the talks

Tim Austin. Irrational L2-Betti numbers

A 1972 question of Atiyah asks for examples of finitely generated groups G together with cocompact free proper G-manifolds whose L2-Betti numbers are irrational. Building on earlier work that converts this into a question about the von Neumann dimension of the kernel of an element of the rational group ring of G, I will describe a recent construction of a family of groups and group ring elements answering this question.

Sylvain Barré. Groups of rank in the interval [1,2]

We will give a definition of the rank of a group acting on a 2-dimensional simplicial complex. Spaces of rank 1 are hyperbolic and on the other hand, the ones of rank 2 are Tits buildings. For example, deleting some building's chambers allows us to construct spaces with rank in ]1,2[. These groups sometimes satisfy property (T), sometimes satisfy the Haagerup property. We will give essentially lot of examples, and we will develop some of their nice properties.

François Dahmani. The isomorphism problem and pinched negative curvature

(Joint work with V. Guirardel) The isomorphism problem, asking for an algorithmic answer to whether two given groups are isomorphic, has recently been solved positively for all hyperbolic groups, and some relatively hyperbolic groups (torsion-free, with abelian parabolic subgroups). The strategy, discovered and first implemented by Sela, relies crucially on the algorithmic solving of equations in these groups, which is a non-trivial step in itself. On the other hand, it is known that there are nilpotent groups in which equations are not algorithmically solvable, so the strategy is put in difficulty for relatively hyperbolic groups with virtually nilpotent parabolic subgroups. The latter class includes fundamental groups of finite volume manifolds with pinched negative curvature. We introduce another method, based on Dehn fillings, which bypasses the procedure of solving equations in these groups, and which goes much beyond this class.

Maria Paula Gomez Aparicio. Twisting property (T) by finite dimensional non-unitary representations

Using its C*-algebraic characterisation, we define a twisting of property (T) which implies that every finite dimensional non-unitary representation ρ of a topological group G is isolated among representations of the form ρ⊗π, where π ranges over the unitary irreducible representations of G. We show that every real simple Lie group having property (T) satisfies the twisted property.

Luc Guyot. Limits of metabelian groups

I will explain how algebraic number theory can be used to describe limits of metabelian groups in the space of marked groups.

Andrei Jaikin Zapirain. Property (T) for groups graded by root systems

I will present a new approach to prove the property (T) for groups graded by root systems and groups associated with graph of groups. The main examples include the elementary linear groups and the Steinberg groups associated with the classical root systems. The talk is based on the joint works with M. Ershov and M. Ershov-M. Kassabov.

Fanny Kassel. Deformation of discrete isometry groups for non-Riemannian symmetric spaces

The deformation of discrete and cocompact groups of isometries of a Riemannian symmetric space is well understood: we know that after a small deformation, such a group still acts properly discontinuously and cocompactly on the symmetric space, and Weil has determined when there are nontrivial small deformations. On the other hand, little is known for non-Riemannian symmetric spaces. In this talk we will show that for a natural class of discrete isometry groups Γ acting properly discontinuously and cocompactly on a non-Riemannian symmetric space, the action of Γ remains properly discontinuous and cocompact after any small deformation. Using the existence of nontrivial deformations of certain arithmetic lattices of SO(1,2n) in SO(2,2n), we will for instance obtain discrete isometry groups of SO(2,2n)/U(1,n) that act properly discontinuously and cocompactly and that are Zariski-dense in SO(2,2n).

Jean Lécureux. Boundary amenability of groups acting on buildings

For a discrete group, to admit a topologically amenable action on a compact space is an interesting property : it is equivalent to Yu's “Property A”, and implies for example that the group satisfies the Novikov conjecture. This property is known to hold for a large class of groups.
After recalling a few basic facts on buildings, I will introduce a new “combinatorial boundary” of buildings, and then explain that the action of the (closed subgroups of the) automorphism group of the building is topologically amenable. This leads to new examples of groups satisfying property A.

Gilbert Levitt. McCool groups

Given a free group Fn, consider the action of Out(Fn) on the set of conjugacy classes of Fn. A McCool group is the stabilizer of a finite subset. I wil explain how to use JSJ theory and outer space to obtain finiteness properties for McCool groups. (joint work with V. Guirardel)

Soyoung Moon. On the class of groups with transitive and faithful amenable actions

We will discuss the class of countable groups admitting an amenable, transitive and faithful action on a countable set. We will study some hereditary properties of this class and in particular we shall show that the double of amenable groups and the amalgamated free products of two amenable groups over a finite subgroup admit such actions.

Assaf Naor. L1-embeddings of the Heisenberg group and fast estimation of graph isoperimetry

We will show that any L1-valued mapping of an ε-net in the unit ball of the Heisenberg group incurs bi-Lipschitz distortion log(1/ε)c, where c is a universal constant. We will also explain how this result implies an exponential improvement to the best known integrality gap for the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.
(Joint work with Jeff Cheeger and Bruce Kleiner)

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