  Let K be a fine hyperbolic graph and G be a group acting on K with finite
quotient. We prove that G is exact provided that all vertex stabilizers are
exact. In particular, a relatively hyperbolic group is exact if all its
peripheral groups are exact. We prove this by showing that the group G acts
amenably on a compact topological space. We include some applications to the
theories of group von Neumann algebras and of measurable orbit equivalence
relations.
