  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 $G$.
