Self-similar tilings and expanding homomorphisms of groups (Q1924574)

We consider self-similar tilings of metric spaces with one tile type, and a group of isometries having a free transitive action on the set of tiles. Pointing out properties of the group endomorphism induced by the expansion of the tiling, we define expanding homomorphisms of groups. We prove that a finitely generated group with an expanding endomorphism has polynomial growth. It is known that self-similar periodic tilings can be encoded by a small amount of algebraic data. The notion of expanding group homomorphisms is applied to the problem to give criteria whether certain algebraic data indeed produce a tiling, and if so, whether this tiling is periodic.