The Borel Distinguishing Number of Schreier Graphs

Abstract

The Borel distinguishing number $D_B(\mathcal{G})$ of a Borel graph $\mathcal{G}$, recently introduced by Bilge and Kaya, is the minimum number of colors required to break the symmetry of $\mathcal{G}$ in a Borel way. In this paper, we investigate the Borel distinguishing number of Schreier graphs induced by the free part of the shift action $\Gamma \curvearrowright n^\Gamma$. We prove that $D_B(\mathcal{G})\le n+1$ for $\Gamma=\mathbb{Z}^d$ equipped with the standard generators. Moreover, we show that $D_B(\mathcal{G})\ge n+1$ if $ \Gamma$ is amenable and $ \{\gamma \in \mathrm{Aut}(\mathrm{Cay}(\Gamma,S)) \mid \gamma(e) = e \}$ is non-trivial. We also show that $D_B(\mathcal{G})$ is finite if $\Gamma$ is finitely generated, and give some applications of our results. These results answer some questions raised by Bilge and Kaya.