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Schedule

Abstracts of the lectures
Damien **Gaboriau** *L*^{2} Betti numbers

The Euler-Poincaré characteristic has been discovered by
Poincaré (1895) to decompose as an alternating sum of meaningful
numbers: the Betti numbers. Atiyah (1976) discovered another
such decomposition as an alternating sum of numbers, which moreover behave
nicely under taking finite index or finite covers:
the L^{2} Betti numbers. They have been largely generalized
(Cheeger-Gromov 1986) so as to encompass group actions and groups.

In the context of discrete group actions on simplicial complexes,
they are defined as generalized dimensions of
the L^{2}-homology (or -cohomology). This generalized
dimension introduced by von Neumann
is defined for Hilbert spaces equipped with a nice action of a group.

The L^{2} Betti numbers of a group *G* form a
sequence (β_{n}(*G*)) of invariants
which are defined using L^{2}-homology (or -cohomology).

I shall present an overview of this subject with a lot of examples and applications.
Pierre **Pansu** *L*^{p}-cohomology and large scale conformal geometry

L^p-cohomology will be defined first for Riemannian manifolds and
simplicial complexes. Examples will be given. Then the definition will be
extended to arbitrary metric spaces. Applications to a large scale notion of
conformal mapping will be given.
Romain **Tessera** L^{p}-*cohomology
of homogeneous Riemannian manifolds*

We will show how the first L^{p}-cohomology
can be used to characterize those
homogeneous Riemannian manifolds that are Gromov hyperbolic. The ideas of
the proof will be illustrated on various explicit examples.

Abstracts of the talks
Matias **Carrasco** Orlicz spaces and quasi-isometries of Heintze groups

We will consider an Orlicz space based cohomology for metric (measured)
spaces with bounded geometry. As for the ordinary power functions, it is a
quasi-isometry invariant of the space. In the hyperbolic case, the degree
one cohomology can be identified with an Orlicz-Besov function space on the
boundary at infinity. By considering a local version of this cohomology, we
give some applications to the large scale geometry of homogeneous spaces
with negative curvature.
Antoine **Gournay** The reduced *l*^{p}-cohomology
in degree 1 and the Poisson boundary

Reduced *l*^{p}-cohomology
in degree 1 (for short "LpR1")
is a useful quasi-isometry invariant of graphs [of bounded valency]
whose definition is relatively simple. On a graph, there is a natural
gradient operator from functions to vertices to functions on edges
defined by looking at the difference of the value on the extremities
of the edge. Simply put, this cohomology is the quotient of functions
with gradient in *l*^{p}
(of the edges) by functions who are
themselves in *l*^{p} (of the vertices).
Jean **Lécureux** Amenable IRS

An Invariant Random Subgroup (IRS) is a measure on the space of
subgroups which is invariant by conjugation. This term was first coined
in a recent paper of Abert, Glasner and Virag, in a paper in which they
raise the question: if the IRS is almost surely amenable, is it
contained in an amenable group ? We answer positively this question. I
will also speak about the case of an IRS contained in a group acting on
a CAT(0) space: if a group acts "geometrically densely" on a CAT(0)
space, then the same is true almost surely of the IRS. This is a joint
work with Uri Bader and Bruno Duchesne for the first part, and Bruno
Duchesne, Yair Glasner and Nir Lazarovich for the second part.
François **Le Maître**
Full groups in the locally compact measure preserving setting

Full groups were introduced in Dye's visionary paper of 1959 as subgroups of
Aut(X,mu) stable under cutting and gluing their elements along a countable
partition of the probability space (X,mu). However, the focus has since then
been rather on full groups of equivalence relations induced by the measure
preserving action of a countable group, which are Polish for the uniform
topology. Indeed, Dye's reconstruction theorem ensures that these full groups,
seen as abstract topological groups, completely determine the equivalence
relation. Then, one can see in some cases how some invariants of the
equivalence relation translate into topological invariants of the full group.
In a work in progress with A. Carderi, we unveil Polish full groups of a new
kind, which arise as the full groups of orbit equivalence relations induced
by probability measure preserving actions of locally compact second
countable groups, and which still satisfy the reconstruction theorem. We will
discuss some of their topological properties such as amenability or topological
rank, that is, the minimal number of elements needed to generate a dense
subgroup.
Henrik **Petersen**
Quasi-isometries of Nilpotent Groups

Every discrete, finitely generated, torsion free nilpotent group G
embeds in a unique connected, simply connected, nilpotent Lie group as a
cocompact lattice. This ambient Lie group is called the Mal'cev completion
of G. A well-known open question asks whether the Mal'cev completion is
invariant under quasi-isometry. In my talk I will present some recent
progress in this direction. This is joint work with David Kyed.
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