自由群作为$3$ -流形的端齐性群
Free groups as end homogeneity groups of $3$-manifolds
摘要：
对于任意的自由群$F$，构造了一个不可约的开$3$ -流形$M_F$，它的端集同胚于Cantor集，并且$M_F$的端齐性群同构于$F$ .端齐性群是端集的所有自同胚的群，这些自同胚扩展到整个$3$ -流形的同胚。这推广了一个较早的结果，即对于每个可生成的阿贝尔群$G$，构造一个不可约的开$3$流形$M_G$，其端部为齐性群$G$ .本文证明的方法还证明了：如果$G$是一个群，其Cayley图在$\mathbb{R}^3$中，且图的自同构具有某些良好的扩张性质，则存在一个不可约的开$3$ -流形$M_G$，其端齐性群为$G$ .
英文摘要：
For every finitely generated free group $F$, we construct an irreducible open $3$-manifold $M_F$ whose end set is homeomorphic to a Cantor set, and with the end homogeneity group of $M_F$ isomorphic to $F$. The end homogeneity group is the group of all self-homeomorphisms of the end set that extend to homeomorphisms of the entire $3$-manifold. This extends an earlier result that constructs, for each finitely generated abelian group $G$, an irreducible open $3$-manifold $M_G$ with end homogeneity group $G$. The method used in the proof of our main result also shows that if $G$ is a group with a Cayley graph in $\mathbb{R}^3$ such that the graph automorphisms have certain nice extension properties, then there is an irreducible open $3$-manifold $M_G$ with end homogeneity group $G$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
