M. Gromov has shown that any two finitely generated groups Γ and Λ are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions Γ y X x Λ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second-countable groups. In his seminal monograph on geometric group theory ...

Minimal, strongly proximal actions of locally compact groups on compact spaces, also known as boundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie group G on homogeneous spaces G/Q where Q ⊂ G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note ...

We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is a finite distance in the sup-norm from a commensurator.

Let ? be a lattice in a simple higher rank Lie group G. We describe all locally compact (not necessarily Lie) groups H in which ? can be embedded as a lattice. For lattices ? in rank one groups G (with the only exception of non-uniform lattices in G ' SL 2 (R), which are virtually free groups) we give a similar description of all possible locally compact groups H, in which ? can be embedded as ...