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arXiv · 2304.14911

Non-concentration property of Patterson-Sullivan measures for Anosov subgroups

Dongryul M. Kim·Hee Oh
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Abstract

Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma<G$ with respect to a parabolic subgroup $P_\theta$, we prove that any $\Gamma$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_\theta$. More generally, we prove that for a Zariski dense $\theta$-transverse subgroup $\Gamma<G$, any $(\Gamma, \psi)$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_\theta$, provided the $\psi$-Poincar\'e series of $\Gamma$ diverges at $s=1$. In particular, our result also applies to relatively Anosov subgroups.

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Dongryul M. Kim, Hee Oh. 2023-04-28. Non-concentration property of Patterson-Sullivan measures for Anosov subgroups. https://arxiv.org/abs/2304.14911

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