Invariant measures for actions of congruent monotileable amenable groups

TL;DR

This paper constructs minimal group actions on subshifts for a broad class of amenable groups, realizing any given Choquet simplex as the space of invariant measures, extending previous results from residually finite groups.

Contribution

It generalizes the existence of invariant measure structures from residually finite to congruent monotileable amenable groups, including virtually nilpotent groups.

Findings

Constructs minimal G-subshifts with prescribed invariant measure spaces.

Extends prior results from residually finite to a larger class of amenable groups.

Includes all infinite countable virtually nilpotent groups.

Abstract

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 \cite{CP} for residually finite groups.