09/21: Wouter van Limbeek
Commensurators and arithmeticity of thin groups
Abstract: The commensurator of a discrete, Zariski-dense subgroup 
of a simple Lie group 
contains information on the arithmetic nature of : For example, in 1974, Margulis proved that if is a lattice and its commensurator is dense, then is arithmetic. In 2011, Shalom asked if the same is true only assuming is Zariski-dense in . I will report on recent progress on this question that uses ideas from infinite ergodic theory, profinite actions, representation theory, Brownian motion, random walks and the structure of locally compact groups. I will show how these combine to give information on commensurators and applications to problems in group theory, hyperbolic 3-manifolds, and structure of arithmetic lattices in Lie groups. This is joint work with D. Fisher and M. Mj.
of a simple Lie group contains information on the arithmetic nature of : For example, in 1974, Margulis proved that if is a lattice and its commensurator is dense, then is arithmetic. In 2011, Shalom asked if the same is true only assuming is Zariski-dense in . I will report on recent progress on this question that uses ideas from infinite ergodic theory, profinite actions, representation theory, Brownian motion, random walks and the structure of locally compact groups. I will show how these combine to give information on commensurators and applications to problems in group theory, hyperbolic 3-manifolds, and structure of arithmetic lattices in Lie groups. This is joint work with D. Fisher and M. Mj.