Tilings of amenable groups

Abstract

We prove that for any infinite countable amenable group $G$, any $\epsilon >0$ and any finite subset $K\subset G$, there exists a tiling (partition of $G$ into finite "tiles" using only finitely many "shapes"), where all the tiles are $(K;\epsilon )$-invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of $G$ (in the sense that the mappings, associated to different from unity elements of $G$, have no fixpoints), on a zero-dimensional space, and which has topological entropy zero.

23 pages

Cited by in corpus (9)

• Nuclear dimension of simple C*-algebras
• Almost finiteness for general etale groupoids and its applications to stable rank of crossed products
• Symbolic dynamics on amenable groups: the entropy of generic shifts
• Minimal subshifts of arbitrary mean topological dimension
• On the entropies of subshifts of finite type on countable amenable groups
• Leptin densities in amenable groups
• Linear repetitivity beyond abelian groups
• On large deviations for amenable group actions
• Clopen type semigroups of actions on $0$-dimensional compact spaces