Invariant Measures for Actions of Congruent Monotileable Amenable Groups

In this paper we show that for every congruent monotileable amenable group G and for every metrizable Choquet simplex K, there exists a minimal G-subshift, which is free on a full measure set, whose set of invariant probability measures is affine homeomorphic to K. If the group is virtually abelian, the subshift is free. Congruent monotileable amenable groups are a generalization of amenable residually finite groups. In particular, we show that this class contains all the infinite countable virtually nilpotent groups. This article is a generalization to congruent monotileable amenable groups of one of the principal results shown in [3] for residually finite groups. © European Mathematical Society.