Schreier图的Borel区分数
The Borel Distinguishing Number of Schreier Graphs
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中文总结 AI 辅助
本文研究Schreier图的Borel区分数，证明了对于$\mathbb{Z}^d$上界为$n+1$，对顺从群给出下界，并证明有限生成时该数有限，回答了Bilge和Kaya的问题。
中文摘要 AI 辅助
Borel图$\mathcal{G}$的Borel区分数$D_B(\mathcal{G})$由Bilge和Kaya最近引入，是指以Borel方式打破$\mathcal{G}$的对称性所需的最少颜色数。本文研究了由移位作用$\Gamma \curvearrowright n^\Gamma$的自由部分诱导的Schreier图的Borel区分数。我们证明当$\Gamma=\mathbb{Z}^d$配备标准生成元时，$D_B(\mathcal{G})\le n+1$。此外，我们证明若$\Gamma$是顺从群且$\{\gamma \in \mathrm{Aut}(\mathrm{Cay}(\Gamma,S)) \mid \gamma(e) = e\}$非平凡，则$D_B(\mathcal{G})\ge n+1$。我们还证明了当$\Gamma$有限生成时$D_B(\mathcal{G})$是有限的，并给出了我们结果的一些应用。这些结果回答了Bilge和Kaya提出的一些问题。
英文摘要
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 $Γ\curvearrowright n^Γ$. We prove that $D_B(\mathcal{G})\le n+1$ for $Γ=\mathbb{Z}^d$ equipped with the standard generators. Moreover, we show that $D_B(\mathcal{G})\ge n+1$ if $ Γ$ is amenable and $ \{γ\in \mathrm{Aut}(\mathrm{Cay}(Γ,S)) \mid γ(e) = e \}$ is non-trivial. We also show that $D_B(\mathcal{G})$ is finite if $Γ$ is finitely generated, and give some applications of our results. These results answer some questions raised by Bilge and Kaya.
