Equivariant maps to subshifts whose points have small stabilizers

TL;DR

This paper extends a result on equivariant maps to subshifts in group actions, showing how to control stabilizers and construct free subshifts with universal properties for free Borel actions.

Contribution

It generalizes prior work by constructing subshifts with prescribed stabilizers, including free subshifts, for a broad class of group actions and subshifts of finite type.

Findings

Existence of subshifts with specified stabilizers for continuous equivariant maps.

Construction of free subshifts within proper colorings of Cayley graphs.

Universal property: every free Borel action admits an equivariant map to these subshifts.

Abstract

Let $\Gamma$ be a countably infinite group. Given $k \in \mathbb{N}$, we use $\mathrm{Free}(k^\Gamma)$ to denote the free part of the Bernoulli shift action of $\Gamma$ on $k^\Gamma$. Seward and Tucker-Drob showed that there exists a free subshift $\mathcal{S} \subseteq \mathrm{Free}(2^\Gamma)$ such that every free Borel action of $\Gamma$ on a Polish space admits a Borel $\Gamma$-equivariant map to $\mathcal{S}$. Here we generalize this result as follows. Let $\mathcal{S}$ be a subshift of finite type (for example, $\mathcal{S}$ could be the set of all proper colorings of the Cayley graph of $\Gamma$ with some finite number of colors). Suppose that $\pi \colon \mathrm{Free}(k^\Gamma) \to \mathcal{S}$ is a continuous $\Gamma$-equivariant map and let $\mathrm{Stab}(\pi)$ be the set of all group elements that fix every point in the image of $\pi$. Unless $\pi$ is constant, $\mathrm{Stab}(\pi)$ is a finite normal subgroup of $\Gamma$. We prove that there exists a subshift $\mathcal{S}' \subseteq \mathcal{S}$ such that the stabilizer of every point in $\mathcal{S}'$ is $\mathrm{Stab}(\pi)$ and every free Borel action of $\Gamma$ on a Polish space admits a Borel $\Gamma$-equivariant map to $\mathcal{S}'$. In particular, if the shift action of $\Gamma$ on the image of $\pi$ is faithful (i.e., if $\mathrm{Stab}(\pi)$ is trivial), then the subshift $\mathcal{S}'$ is free. As an application of this general result, we deduce that if $F$ is a finite symmetric subset of $\Gamma \setminus \{\mathbf{1}\}$ of size $|F| = d \geq 1$ and $\mathrm{Col}(F, d + 1) \subseteq (d+1)^\Gamma$ is the set of all proper $(d+1)$-colorings of the Cayley graph of $\Gamma$ corresponding to $F$, then there is a free subshift $\mathcal{S} \subseteq \mathrm{Col}(F, d+1)$ such that every free Borel action of $\Gamma$ on a Polish space admits a Borel $\Gamma$-equivariant map to $\mathcal{S}$.