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

Ilya Kapovich
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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 $G$.

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Ilya Kapovich. 2002-03-03. The non-amenability of Schreier graphs for infinite index quasiconvex subgroups of hyperbolic groups. https://arxiv.org/abs/math/0201076

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