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arXiv · 2408.11274

Exponential prime orbit theorems for Anosov subgroups

Michael Chow·Pratyush Sarkar
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Abstract

Let $\Gamma$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group -- these are higher rank analogues of convex cocompact subgroups. Let us measure the Jordan projections with any linear form which is positive on the limit cone of $\Gamma$. We prove a corresponding counting theorem with a power saving error term for the conjugacy classes of loxodromic elements in $\Gamma$. The proof is based on interpreting the Jordan projections as periods of a natural flow associated to $\Gamma$ and proving exponential mixing. We also prove the existence of a spectral gap for the Selberg zeta function.

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Michael Chow, Pratyush Sarkar. 2024-08-21. Exponential prime orbit theorems for Anosov subgroups. https://arxiv.org/abs/2408.11274

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