Tilings of amenable groups

TL;DR

This paper proves the existence of invariant tilings in amenable groups with zero entropy and constructs free actions with zero entropy, advancing understanding of group tilings and dynamical systems.

Contribution

It introduces a method to tile amenable groups with invariant tiles and constructs free zero-entropy actions on zero-dimensional spaces.

Findings

Existence of $(K; \epsilon)$-invariant tilings for amenable groups.

Construction of free actions with topological entropy zero.

Tilings have subexponential complexity of patterns.

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.