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arXiv · 2404.09745

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

Dongryul M. Kim·Hee Oh
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Abstract

For a geometrically finite Kleinian group $\Gamma$, 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-Peign\'e 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 $\Gamma$ 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 $\Gamma$ 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.

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Dongryul M. Kim, Hee Oh. 2024-04-15. Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization. https://arxiv.org/abs/2404.09745

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