We characterise hyperbolic groups as uniform convergence groups. More precisely, we show the following. Suppose $ M $ is a perfect compact metrisable space, and suppose that a group, $ \Gamma $, acts by homeomorphism on $ M $ such that the induced action on the space of disctinct triples of $ M $ is properly discontinuous and cocompact. Then $ \Gamma $ is hyperbolic, and $ M $ is equivariantly homeomorphic to the boundary of $ \Gamma $. We also give a variation on this charactersation, by showing that it is sufficient that $ \Gamma $ should act as a convergence group on $ M $ (i.e. properly discontinuously on the space of distinct triples) such that every point of $ M $ is a conical limit point.
J. Amer. Math. Soc. 11 (1998) 643-667.