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# Continuous Combinatorics of Abelian Group Actions

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### Abstract

This paper develops techniques which are used to answer a number of questions in the theory of equivalence relations generated by continuous actions of abelian groups. The methods center around the construction of certain specialized hyper-aperiodic elements, which produce compact subflows with useful properties. For example, we show that there is no continuous $$3$$-coloring of the Cayley graph on $$F(2^{\mathbb{Z}^2})$$, the free part of the shift action of $$\mathbb{Z}^2$$ on $$2^{\mathbb{Z}^2}$$. With earlier work of the authors this computes the continuous chromatic number of $$F(2^{\mathbb{Z}^2})$$ to be exactly $$4$$. Combined with marker arguments for the positive directions, our methods allow us to analyze continuous homomorphisms into graphs, and more generally equivariant maps into subshifts of finite type. We present a general construction of a finite set of "tiles" for $$2^{\mathbb{Z}^n}$$ (there are $$12$$ for $$n=2$$) such that questions about the existence of continuous homomorphisms into various structures reduce to finitary combinatorial questions about the tiles. This tile analysis is used to deduce a number of results about $$F(2^{\mathbb{Z}^n})$$.

### Most cited references6

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### Borel Chromatic Numbers

(1999)
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### Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. I

(1977)
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### Countable abelian group actions and hyperfinite equivalence relations

(2015)
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### Author and article information

###### Journal
10 March 2018
###### Article
1803.03872