Discrete Linear Groups containing Arithmetic Groups

Abstract

We prove in a large number of cases, that a Zariski dense discrete subgroup of a simple real algebraic group $G$ which contains a higher rank lattice is a lattice in the group $G$. For example, we show that a Zariski dense subgroup of $S{L}_n(\mathbb{R})$ which contains $S{L}_3(\mathbb{Z})$ in the top left hand corner, is conjugate to $S{L}_n(\mathbb{Z})$ .

Accepted by Journal de l'Ecole Polytechnique