The fundamental inequality for cocompact Fuchsian groups

Abstract

We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.

Invariant measures of the topological flow and measures at infinity on hyperbolic groups

“…Therefore if τ(ρ/d S ) ≠ τ(ρ/d S ∶ Λ harm ), then a Patterson-Sullivan measure µ S and the harmonic measure ν are mutually singular on ∂Γ; the fact itself has been shown as a special case of [28, Theorems 1.2 and 1.3]. Moreover, we note that µ ρ and ν are mutually singular[35, Theorem 1]; see also the discussion in Remark 6.2.…”

“…In particular, this applies to finite range random walks on surface groups. In certain special cases, it has been recently shown that the harmonic measure is singular with respect to the Lebesgue measure on the boundary of the hyperbolic plane [34, Theorem 1.1] and [35,Theorem 1]. In those cases, θ G, d is strictly convex.…”

Asymptotics of the first-passage function on free and Fuchsian groups

“…In [Kos20] and [KT22] we developed techniques that allowed us to confirm Conjecture 1.1 for nearestneighbour random walks on cocompact Fuchsian groups Γ generated by side-pairing transformations (t 1 , . .…”

Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups

“…In [CLP21] and [Kos21] the conjecture is verified for nearest neighbor random walks on tilings by regular polygons. A general result for symmetric random walks on the fundamental group of the surface of genus two is obtained in [KT22]). For any finite support probability measure on the fundamental group of a closed surface, dimension drop for the harmonic measure associated to discrete and faithful representations of the group outside of a compact subset of Teichmüller space was established in [AGG + 22].…”