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arXiv · 2607.13730

Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups

Eduardo Reyes·Tianqi Wang
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Abstract

We establish several results about Patterson--Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves--Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are $C^1$-regular, from which we deduce that the growth indicator is $C^1$-regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim--Oh--Wang. Our methods are dynamical, and we exploit the fact due to Kim--Oh and Blayac--Canary--Zhu--Zimmer that Bowen--Margulis--Sullivan measures for relatively Anosov groups are finite and mixing.

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Eduardo Reyes, Tianqi Wang. 2026-07-15. Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups. https://arxiv.org/abs/2607.13730

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