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Abstracts of the lectures

Damien Gaboriau L2 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 L2 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 L2-homology (or -cohomology). This generalized dimension introduced by von Neumann is defined for Hilbert spaces equipped with a nice action of a group.
      The L2 Betti numbers of a group G form a sequence (βn(G)) of invariants which are defined using L2-homology (or -cohomology).
     I shall present an overview of this subject with a lot of examples and applications.

Pierre Pansu Lp-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 Lp-cohomology of homogeneous Riemannian manifolds

We will show how the first Lp-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 lp-cohomology in degree 1 and the Poisson boundary

Reduced lp-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 lp (of the edges) by functions who are themselves in lp (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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