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

Frączyk, Mikołaj; Lee, Minju

doi:10.2140/gt.2025.29.1017

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

Mikołaj Frączyk and Minju Lee

Geometry & Topology 29 (2025) 1017–1036

DOI: 10.2140/gt.2025.29.1017

Abstract

Let $G$ be a connected semisimple real algebraic group and $Γ < G$ be a Zariski dense discrete subgroup. We prove that if $Γ ∖ G$ 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 the classification of convex cocompact actions by Kleiner and Leeb (Invent. Math. 163 (2006) 657–676) and Quint (Geom. Dedicata 113 (2005) 1–19), 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 and Lindenstrauss (Ann. of Math. 164 (2006) 513–560). In a companion paper jointly with Edwards and Oh (C. R. Math. Acad. Sci. Paris 362 (2024) 1873–1880) 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.

Keywords

Bowen–Margulis–Sullivan measure, higher rank

Mathematical Subject Classification

Primary: 37A17

References

Publication

Received: 18 July 2023

Revised: 23 March 2024

Accepted: 30 May 2024

Published: 21 April 2025

Proposed: David Fisher

Seconded: Benson Farb, Mladen Bestvina

Authors

Mikołaj Frączyk
Faculty of Mathematics and Computer Science Jagiellonian University Krakow Poland

Minju Lee
Department of Mathematics University of Chicago Chicago, IL United States

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Volume 29, issue 2 (2025)