Irreducible groups and ergodicity in the boundary

Abstract

We show that if $G$ is a real semi-simple Lie group, and $\Gamma$ is a discrete subgroup of $G$ containing a subgroup $\Sigma$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $\Gamma$ is not isomorphic to a free product of $\Sigma$ with $\mathbb{Z}$. Moreover, if $\Sigma$ has algebraic entries, then $\Gamma$ has algebraic entries as well. As a consequence, we show that if all irreducible discrete subgroups of ${\rm SL}_2(\mathbb{R}) \times {\rm SL}_2(\mathbb{R})$ act ergodically on $\mathbb{S}^1\times \mathbb{S}^1 $, such groups cannot be free groups (or even Gromov hyperbolic). In the appendix, we discuss a connection between the existence of discrete irreducible groups and diophantine properties of Lie groups.