Classifying lattice envelopes of some countable groups.

In a joint work with Uri Bader and Roman Sauer we study the following question:
given a countable group L describe all locally compact groups G which
contain a copy of L as a lattice (uniform or non-uniform).

I will discuss the solution of this problem for a large class of
countable groups L. The proof involves a somewhat unexpected mix of tools, including:
Breuillard-Gelander’s topological Tits alternative, Margulis’
commensurator superrigidity, arithmeticity, and normal subgroup
theorems, quasi-isometric rigidity results of Kleiner-Leeb, and
Mosher-Sageev-Whyte.