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

Frączyk, Mikołaj¹ ; Lee, Minju²

¹Faculty of Mathematics and Computer Science, Jagiellonian University, Krakow, Poland
²Department of Mathematics, University of Chicago, Chicago, IL, United States

Geometry & topology, Tome 29 (2025) no. 2, pp. 1017-1036

Résumé

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 L2 spectrum has no atom in any infinite volume quotient of a higher rank simple algebraic group.

Détail

DOI : 10.2140/gt.2025.29.1017

Keywords: Bowen–Margulis–Sullivan measure, higher rank

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