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

We consider the harmonic measure on the Gromov boundary of a non-amenable 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 show...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 163; no. 1; pp. 285 - 316
Main Authors Izumi, Masaki, Neshveyev, Sergey, Okayasu, Rui
Format Journal Article
LanguageEnglish
Published New York Springer-Verlag 2008
Subjects
Algebra
Analysis
Applications of Mathematics
Group Theory and Generalizations
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Theoretical
Gibbs Measure
Random Walk
Hyperbolic Group
Harnack Inequality
Harmonic Measure
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Summary:We consider the harmonic measure on the Gromov boundary of a non-amenable 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 III 0 .
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-008-0013-6