Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
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Dongryul M. Kim
und
Hee Oh
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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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence
Artikel in diesem Heft
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence
