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arXiv · 1502.02413

Tilings of amenable groups

Tomasz Downarowicz·Dawid Huczek·Guohua Zhang
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Abstract

We prove that for any infinite countable amenable group $G$, any $ε> 0$ 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 $(K; ε)$-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.

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Tomasz Downarowicz, Dawid Huczek, Guohua Zhang. 2015-02-09. Tilings of amenable groups. https://arxiv.org/abs/1502.02413

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