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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 *H*∫_{X}*G* 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 L*^{2}-Betti numbers

A 1972 question of Atiyah asks for examples of finitely generated
groups *G* together with cocompact free proper *G*-manifolds
whose
L^{2}-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(1,2*n*) in SO(2,2*n*), we will for
instance obtain discrete isometry groups of
SO(2,2*n*)/U(1,*n*) that act properly discontinuously and
cocompactly and that are Zariski-dense in SO(2,2*n*).
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 *F*_{n}, consider the action of
Out(*F*_{n}) on the set of
conjugacy classes of *F*_{n}. 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**. *L*^{1}-embeddings of the Heisenberg
group and fast estimation of graph isoperimetry

We will show
that any L^{1}-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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