Continuous Combinatorics of Abelian Group Actions

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 \(F(2^{\mathbb{Z}^n})\).