The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups

Abstract

We show that if $H$ is a quasiconvex subgroup of infinite index in a non-elementary hyperbolic group $G$ then the Schreier coset graph $X$ for $G$ relative to $H$ is non-amenable (that is, $X$ has positive Cheeger constant). We present some corollaries regading the Martin boundary and Martin compactification of $X$ and the co-growth of $H$ in $G$.

arXiv commentupdated version

References in corpus (2)

• Percolation Perturbations in Potential Theory and Random Walks
• The geometry of relative Cayley graphs for subgroups of hyperbolic groups

Cited by in corpus (6)

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• Multiplicative measures on free groups
• Generic-case complexity, decision problems in group theory and random walks