The above definition of convergence group admits a useful equivalent reformulation in terms of the action of \Gamma on the ``space of distinct triples" of M. For a set M denote \Theta(M):=M^3\setminus \Delta(M), where \Delta(M)=\{(a,b,c)\in M^3\mid \#\{a,b,c\}\le 2\}.
Let \Gamma be a group acting by homeomorphisms on a compact metrizable space M with at least two points.