Irreducible groups and ergodicity in the boundary

Abstract

We show that if $G$ is a real semi-simple Lie group, and $\hat{I}“$ is a discrete subgroup of $G$ containing a subgroup $\hat{I}\pounds$ acting ergodically (in a strong sense) on the Furstenberg boundary of $G$, then $\hat{I}“$ is not isomorphic to a free product of $\hat{I}\pounds$ with $\mathbb{Z}$. Moreover, if $\hat{I}\pounds$ has algebraic entries, then $\hat{I}“$ has algebraic entries as well. As a consequence, we show that if all irreducible discrete subgroups of ${\mathrm{SL}}_2(\mathbb{R})\times {\mathrm{SL}}_2(\mathbb{R})$ 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.