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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. [ABSTRACT FROM AUTHOR]

We provide a general criterion to deduce maximal amenability of von Neumann subalgebras LΛ ⊂ LΓ arising from amenable subgroups Λ of discrete countable groups Γ. The criterion is expressed in terms of Λ-invariant measures on some compact Γ-space. The strategy of proof is different from Popa's approach to maximal amenability via central sequences (Adv Math 50:27-48, ), and relies on elementary computations in a crossed-product C-algebra. [ABSTRACT FROM AUTHOR]