Some results and questions concerning lattices in totally disconnected groups

14 December 2012
Tsachik Gelander

I'll discuss some results about lattices in totally
disconnected locally compact groups, elaborating on the question:
which classical results for lattices in Lie groups can be extended to
general locally compact groups. For example, in contrast to Borel's
theorem that every simple Lie group admits (many) uniform and
non-uniform lattices, there are totally disconnected simple groups
with no lattices. Another example concerns with the theorem of Mostow
that lattices in connected solvable Lie groups are always uniform.
This theorem cannot be extended for general locally compact groups,
but variants of it hold if one implants sufficient assumptions. At
least 90% of what I intend to say is taken from a paper and an
unpublished preprint written jointly with P.E. Caprace, U. Bader and
S. Mozes. If time allows, I will also discuss some basic properties
and questions regarding Invariant Random Subgroups.