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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

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Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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1435-5345
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Volume 2019, Issue 747

# Tilings of amenable groups

Tomasz Downarowicz
• Institute of Mathematics of the Polish Academy of Science, Śniadeckich 8, 00-956 Warszawa, Poland; and Faculty of Pure and Applied Mathematics, Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
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/ Dawid Huczek
• Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
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• Other articles by this author:
/ Guohua Zhang
• School of Mathematical Sciences and LMNS, Fudan University, and Shanghai Center for Mathematical Sciences, Shanghai 200433, P. R. China
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• Other articles by this author:
Published Online: 2016-07-16 | DOI: https://doi.org/10.1515/crelle-2016-0025

## 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 $\left(K,\epsilon \right)$-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 elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

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Revised: 2016-02-13

Published Online: 2016-07-16

Published in Print: 2019-02-01

Funding Source: Narodowe Centrum Nauki

Award identifier / Grant number: 2013/08/A/ST1/00275

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11271078

The research of the first two authors is funded by NCN grant 2013/08/A/ST1/00275. Guohua Zhang is supported by NSFC (11271078).

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 747, Pages 277–298, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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