A group $\Gamma$ is said to be “finitely non-co-Hopfian,” or “renormalizable,” if there exists a self-embedding $\varphi \colon \Gamma \to \Gamma$ whose image is a proper subgroup of finite index. Such a proper self-embedding is called a “renormalization for $\Gamma$.” In this work, we associate a dynamical system to a renormalization $\varphi$ of $\Gamma$. The discriminant invariant ${\mathcal D}_{\varphi}$ of the associated Cantor dynamical system is a profinite group which is a measure of the asymmetries of the dynamical system. If ${\mathcal D}_{\varphi}$ is a finite group for some renormalization, we show that $\Gamma/C_{\varphi}$ is virtually nilpotent, where $C_{\varphi}$ is the kernel of the action map. We introduce the notion of a (virtually) renormalizable Cantor action, and show that the action associated to a renormalizable group is virtually renormalizable. We study the properties of virtually renormalizable Cantor actions, and show that virtual renormalizability is an invariant of continuous orbit equivalence. Moreover, the discriminant invariant of a renormalizable Cantor action is an invariant of continuous orbit equivalence. Finally, the notion of a renormalizable Cantor action is related to the notion of a self-replicating group of automorphisms of a rooted tree.

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可重整群的康托动力学

如果存在自嵌入 $\varphi \colon \Gamma \to \Gamma$ 其图像是有限索引。这种适当的自嵌入被称为“$\Gamma$ 的重整化”。在这项工作中，我们将动态系统与 $\Gamma$ 的重整化 $\varphi$ 相关联。相关康托动力系统的判别不变式 ${\mathcal D}_{\varphi}$ 是一个超限群，它是动力系统不对称性的度量。如果 ${\mathcal D}_{\varphi}$ 是一些重整化的有限群，我们证明 $\Gamma/C_{\varphi}$ 几乎是幂零的，其中 $C_{\varphi}$ 是行动地图。我们引入了（实际上）可重整化的康托尔动作的概念，并证明与可重整化群相关的动作实际上是可重整化的。我们研究了虚拟可重整康托作用的性质，并表明虚拟可重整性是连续轨道等价的不变量。此外，可重整康托作用的判别式不变量是连续轨道等价的不变量。最后，可重整康托动作的概念与有根树的自同构自复制群的概念有关。