Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups

We prove that for any infinite, maximal amenable subgroup $H$ in a hyperbolic group $G$, the von Neumann subalgebra $LH$ is maximal amenable inside $LG$. It provides many new, explicit examples of maximal amenable subalgebras in II$_1$ factors. We also prove similar maximal amenability results for direct products of relatively hyperbolic groups and orbit equivalence relations arising from measure-preserving actions of such groups.
Comment: 16 pages, 2 figures, new section added about direct products. Accepted for publication to Mathematische Annalen