Irreducible groups and ergodicity in the boundary

TL;DR

This paper investigates the structure of discrete subgroups in semi-simple Lie groups, showing ergodic actions prevent certain free product decompositions and relate algebraic properties to boundary actions.

Contribution

It establishes that ergodic actions on the Furstenberg boundary impose restrictions on subgroup decompositions and algebraic entries, extending to specific cases like SL(2,R)×SL(2,R).

Findings

Ergodic subgroups cannot be free products with Z.

Subgroups with algebraic entries preserve algebraic properties.

Certain irreducible groups in SL(2,R)×SL(2,R) cannot be free or hyperbolic.

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.