14.00 – 14.45 : Jean Lécureux (Université de Bordeaux)

Title: A simple lattice in an affine building

Abstract: Affine buildings appear naturally in the study of algebraic groups over local fields: the first example is the tree on which acts SL_2(Q_p). Lattices in these algebraic groups are, in higher rank, arithmetic, and in every case, residually finite (hence never simple).

Nevertheless, the main result that I will present implies that there exists a simple group which acts properly discontinuously and cocompactly on an affine building. The construction of this group is due to Titz-Mite and Witzel, and together with Witzel we are able to conclude to its simplicity. The main tool we use is a construction of an analogue of a « geodesic flow » on the building, with an adequate measure, and prove its ergodicity. I will try to present some ideas for these constructions, focusing on the example of the tree. 

15.00 – 15.45 : Adrien Le Boudec (CNRS & ENS de Lyon) 

Title: Solvable groups with a common cocompact envelope

Abstract: A locally compact group $G$ is a cocompact envelope of a group $\Gamma$ if $G$ contains a copy of $\Gamma$ as a discrete and cocompact subgroup. We consider the problem that takes two finitely generated groups having a common cocompact envelope and asks what properties must be shared between them, for the class of solvable groups of finite rank. In that setting we obtain both rigidity and flexibility results. We obtain in particular that the class of solvable groups of finite rank is not QI-rigid. Our flexibility results also allow for finitely presented groups, and more generally groups with type $F_n$ for arbitrary $n$.

16.15 – 17.00 : Vadim Kaimanovich (Université de Rennes)

Title: Collapsing harmonic measures for discrete subgroups of semisimple Lie groups

Abstract: The Furstenberg boundary of a non-compact, finite centre, real semi-simple Lie group (equivalently, of the associated Riemannian symmetric space) is its quotient by a minimal parabolic subgroup; for $SL(d,\mathbb R)$ this is the complete flag variety in $\mathbb R^d$. It serves as a « skeleton » of various compactifications and is essential for understanding the large-scale geometry of the symmetric space.

It is known since the 1980s that, under natural conditions, a random walk on a discrete subgroup gives rise to a uniquely defined harmonic measure on the Furstenberg boundary. This measure makes the boundary isomorphic, as a measure space, to the Poisson boundary of the random walk.

There is also a canonical finite family of lower dimensional quotients of the Furstenberg boundary corresponding to non-minimal parabolic subgroups (partial flag varieties in the $SL(d,\mathbb R)$ case). In general – for instance, when the harmonic measure is absolutely continuous – these quotient maps yield non-trivial quotients of the Poisson boundary.

The purpose of the talk is to exhibit a new « collapsing » phenomenon: there are situations in which some of these quotient maps become measure-theoretic isomorphisms with respect to the harmonic measure. In such cases the Poisson boundary is effectively « smaller » than the full geometric Furstenberg boundary. The construction uses the work of Hochman- Solomyak on the dimension of the harmonic measure for countable,non-discrete groups of isometries of the hyperbolic plane.