Boundary amenability of hyperbolic spaces

It is well-known that a Kleinian group is amenable if and only if it is elementary. We establish an analogous property for equivalence relations and foliations with Gromov hyperbolic leaves: they are amenable if and only if they are elementary in the sense that one can assign (in a measurable way) to any leaf a finite subset of its hyperbolic boundary (as in the group case, such subsets cannot actually contain more than 2 points). The analogous result for actions of word hyperbolic groups with a quasi-invariant measure is that such an action is amenable if and only if it factorizes through the hyperbolic boundary or its symmetric square. A byproduct of our approach is a proof of boundary amenability for isometry groups of Gromov hyperbolic spaces under the assumption that either the space is exponentially bounded or it is CAT(−1) and the group has a finite critical exponent. We also give examples showing that without these assumptions the boundary amenability may fail.