The complex of cuts in a Stone space

Stone’s representation theorem asserts a duality between Boolean algebras on the one hand and Stone spaces , which are compact, Hausdorff and totally disconnected, on the other. This duality implies a natural isomorphism between the homeomorphism group of the space and the automorphism group of the algebra. We introduce a complex of cuts on which these groups act and prove that when the algebra is countable and the space has at least five points, these groups are the full automorphism group of the complex.