  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.
