Continuous Combinatorics of Abelian Group Actions

Source 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, free subflows with useful properties. For example, we show that there is no continuous proper 3 3 -coloring of the Schreier graph on F ( 2 Z 2 ) F(2^{\mathbb {Z}^2}) , the free part of the shift action of Z 2 \mathbb {Z}^2 on 2 Z 2 2^{\mathbb {Z}^2} . With earlier work of Gao and Jackson (2015) this computes the continuous chromatic number of F ( 2 Z 2 ) F(2^{\mathbb {Z}^2}) to be exactly 4 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 Z n 2^{\mathbb {Z}^n} (there are 12 12 for n = 2 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 Z n ) F(2^{\mathbb {Z}^n}) .