The fundamental inequality for cocompact Fuchsian groups
Petr Kosenko, Giulio Tiozzo

TL;DR
This paper demonstrates that for certain hyperbolic groups, the hitting measure of random walks is singular and has Hausdorff dimension less than one, revealing deep geometric and measure-theoretic properties.
Contribution
It establishes the singularity and dimension bounds of hitting measures for cocompact Fuchsian groups and related Coxeter groups, introducing a new geometric inequality for geodesic lengths.
Findings
Hitting measure is singular with respect to Lebesgue measure.
Hausdorff dimension of the hitting measure is less than 1.
A new geometric inequality for geodesic lengths is proven.
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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Coordination Chemistry and Organometallics
