标题： 关于双可逆自动机和群在它们的Cayley图自同构中的共轭者 显示英文标题

标题： On bireversible automata and commensurators of groups in automorphisms of their Cayley graphs

摘要： 如果$G$是一个有限生成群，$X$是$G$的一个Cayley图，用$\mathcal{C}_1^X(G)$表示所有与$G$保持共轭并固定对应于单位元的顶点的$X$的自同构子群。在Macedońska、Nekrashevych和Sushchansky工作的基础上，我们注意到$\mathcal{C}_1^X(G)$可以表示为由双可逆自动机生成的群的有向并集。 我们用此来证明$\mathcal{C}_1^X(G)$的每个循环子群都是无扭曲的，并得到$G$的一个必要条件，使得$\mathcal{C}_1^X(G)$不是局部有限的。 作为推论，我们证明了几类群不能由双可逆自动机生成，并表明由双可逆自动机生成的群的集合严格包含在由可逆和可逆自动机生成的群的集合中。 显示英文摘要

摘要： If $G$ is a finitely generated group and $X$ is a Cayley graph of $G$, denote by $\mathcal{C}_1^X(G)$ the subgroup of all automorphisms of $X$ commensurating $G$ and fixing the vertex corresponding to the identity. Building on the work of Macedo\'{n}ska, Nekrashevych and Sushchansky, we observe that $\mathcal{C}_1^X(G)$ can be expressed as a directed union of groups generated by bireversible automata. We use this to to show that every cyclic subgroup of $\mathcal{C}_1^X(G)$ is undistorted and to obtain a necessary condition on $G$ for $\mathcal{C}_1^X(G)$ not to be locally finite. As a consequence, we prove that several families of groups cannot be generated by bireversible automata and show that the set of groups generated by bireversible automata is strictly contained in the set of groups generated by invertible and reversible automata.