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

TL;DR

The paper addresses when a discrete subgroup $\Gamma$ of a higher-rank semisimple group $G$ supports a finite Bowen-Margulis-Sullivan measure. It develops a measure-theoretic framework using Patterson–Sullivan conformal measures, leafwise measures, and the high-entropy rigidity method of EKL to show that a finite BMS measure forces $\Gamma$ to be virtually a product of a higher-rank lattice and rank-one subgroups corresponding to rank-one factors of $G$. This yields a rigidity-type classification mirroring convex cocompact actions in higher rank and implies that absence of rank-one factors forces $\Gamma$ to be a lattice in $G$. The results connect dynamics of $A$-actions with algebraic structure, providing a tool to identify lattices via finiteness of the BMS measure and linking to spectral consequences in related work.

Abstract

Let $G$ be a connected semisimple real algebraic group and $Γ<G$ be its Zariski dense discrete subgroup. We prove that if $Γ\backslash G$ admits any finite Bowen-Margulis-Sullivan measure, then $Γ$ 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.

Theorems & Definitions (45)

• Theorem 1.1: Kleiner-Leeb
• Remark 1.2
• Theorem 1.3: Quint
• Definition 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8: Edwards-Fraczyk-Lee-Oh EFLO
• Proposition 2.6: Quint
• Lemma 2.9
• Lemma 2.10