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arXiv · 2012.07417

The fundamental inequality for cocompact Fuchsian groups

Petr Kosenko·Giulio Tiozzo
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Abstract

We prove that the hitting measure is singular with respect to Lebesgue measure for any random walk on a cocompact Fuchsian group generated by translations joining opposite sides of a symmetric hyperbolic polygon. Moreover, the Hausdorff dimension of the hitting measure is strictly less than 1. A similar statement is proven for Coxeter groups. Along the way, we prove for cocompact Fuchsian groups a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.

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Petr Kosenko, Giulio Tiozzo. 2020-12-14. The fundamental inequality for cocompact Fuchsian groups. https://arxiv.org/abs/2012.07417

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