Rigidity of Homomorphisms from Arithmetic Groups to Totally Disconnected Groups

Abstract

The talk will explain joint work with Yehuda Shalom showing that the only homomorphisms from certain arithmetic groups to totally disconnected, locally compact groups are the obvious, or naturally occurring, ones. For these groups, this extends the superrigidity theorem that G. Margulis proved for homomorphisms from high rank arithmetic groups to Lie groups. The theorems will be illustrated by referring to the groups \(SL_3(\mathbb{Z})\), \(SL_2(\mathbb{Z}[\sqrt{2}])\) and \(SL_3(\mathbb{Q})\).