Mean topological dimension for actions of discrete amenable groups

Abstract

Let $G$ be a countable amenable group containing subgroups of arbitrarily large finite index. Given a polyhedron $P$ and a real number $\rho$ such that $0 \leq \rho \leq$dim$(P)$, we construct a closed subshift $X \subset P^G$ having mean topological dimension $\rho$. This shows in particular that mean topological dimension of compact metrisable $G$-spaces take all values in $[0,\infty]$.