Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization

Dongryul M. Kim; Hee Oh

doi:10.1515/crelle-2025-0040

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Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization

Dongryul M. Kim and Hee Oh

Published/Copyright: July 3, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 826

Abstract

For a geometrically finite Kleinian group Γ, the Bowen–Margulis–Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal–Peigné respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup Γ of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves–Manning space of Γ which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen–Margulis–Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1900101

Funding statement: Hee Oh is partially supported by the NSF grant No. DMS-1900101.

Acknowledgements

We were informed that the ongoing work of Blayac, Canary, Zhu and Zimmer [4] contains a different proof of Theorem 1.1.

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Received: 2024-05-16

Revised: 2025-05-16

Published Online: 2025-07-03

Published in Print: 2025-09-01

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