Discrete subgroups with finite Bowen-Margulis-Sullivan measure in higher rank

TL;DR

This paper characterizes when a discrete subgroup of a higher rank semisimple algebraic group admits a finite Bowen-Margulis-Sullivan measure, revealing a structure akin to higher rank lattices and rank one factors, with implications for spectral theory.

Contribution

It proves that such subgroups are virtually products of higher rank lattices and rank one factors, extending measure classification in higher rank groups.

Findings

Finite Bowen-Margulis-Sullivan measure implies the subgroup's product structure.

Subgroups are virtually products of higher rank lattices and rank one factors.

Application to spectral theory showing no atom at the bottom of the $L^2$ spectrum.

Abstract

Let $G$ be a connected semisimple real algebraic group and $\Gamma<G$ be its Zariski dense discrete subgroup. We prove that if $\Gamma\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $\Gamma$ is virtually a product of higher rank lattices and discrete subgroups of rank one factors of $G$. This may be viewed as a measure-theoretic analogue of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. The key ingredients in our proof are the product structure of leafwise measures and the high entropy method of Einsiedler-Katok-Lindenstrauss. In a companion paper jointly with Edwards and Oh, we use this result to show that the bottom of the $L^2$ spectrum has no atom in any infinite volume quotient of a higher rank simple algebraic group.