论文标题
免费组作为$ 3 $ manifolds的最终同质性集团
Free groups as end homogeneity groups of $3$-manifolds
论文作者
论文摘要
对于每个有限生成的免费组$ f $，我们构建了一个不可约的开放$ 3 $ -manifold $ m_f $，其最终设置是同型cantor set，并且最终同质性组为$ m_f $ isomorphic to $ f $。最终同质性小组是终点集的所有自我塑料的组，该组扩展到整个$ 3 $ manifold的同构。这扩展了较早的结果，该结果构建了，对于每个有限生成的Abelian Group $ G $，不可约开放的$ 3 $ -MANIFOLD $ M_G $，以及END同质性组$ G $。在我们主要结果证明中使用的方法还表明，如果$ g $是$ \ mathbb {r}^3 $中的Cayley图的组，使得图形自动形态具有某些不错的扩展属性，那么有一个不可约合的开放$ 3 $ manifold $ m_g $ m_g $，带有最终同型组$ g $。
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 $G$.
