The fundamental inequality for cocompact Fuchsian groups

2020/12/14 by Petr Kosenko, Kosenko, Petr, Giulio Tiozzo +1 · 3 citations
Chemistry · Mathematics · #20F67 (Primary) 30F35 #60G50 #60J50 (Secondary) #Coordination Chemistry and Organometallics #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2012.07417

openalex publication_date 2020/12/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Random walks on cocompact Fuchsian and Kleinian groups
• Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
• A note on the singularity conjecture for infinite covolume discrete subgroups

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