The ratio set of the harmonic measure of a random walk on a hyperbolic group

2006/02/19 by Izumi, Masaki, Neshveyev, Sergey, Okayasu, Rui
#Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Probability (math.PR)

paper · doi:10.48550/arxiv.math/0602409

Abstract

We consider the harmonic measure on the Gromov boundary of a nonamenable hyperbolic group defined by a finite range random walk on the group, and study the corresponding orbit equivalence relation on the boundary. It is known to be always amenable and of type III. We determine its ratio set by showing that it is generated by certain values of the Martin kernel. In particular, we show that the equivalence relation is never of type III₀.

Related

• Dimensional properties of the harmonic measure for a random walk on a hyperbolic group
• The Hausdorff dimension of the harmonic measure for relatively hyperbolic groups
• Martin boundary covers Floyd boundary
• Local limit theorem for symmetric random walks in Gromov-hyperbolic groups
• Ratio-limit boundaries for random walks on relatively hyperbolic groups