论文标题
限制亚组的换向引理，并应用于作用于根部树上的组
A commutator lemma for confined subgroups and applications to groups acting on rooted trees
论文作者
论文摘要
如果$ h $的$ g $ h $在space $ \ operatotorname {sub}（sub}（g）的$ g $的子组中，则限制了一个组$ g $的子组$ h $。我们证明了一个换向子组的换向器引理。对于同构同构的组，这为正常亚组的经典换向器引理提供了确切的类似物（尤其是urss）：如果$ g $是一组同构的同态同态的hausdorff space $ x $，$ h $是$ g $的$ g $的限制子，那么$ g $的开放式是，$ h $ contrive in Derive nifive nirive of derive nirive of der nifive of derive of der of der nifiend of der of der的限制。 $ x $。我们将此换向器的引理应用于作用于植根树的组。我们证明了一个定理，描述了弱分支组的URS及其非本质上的最小作用的结构。在这些结果的应用中，我们显示：1）如果$ g $是有限生成的分支组，则$ G $ - 部分T $的$ G $ ACTION在所有忠实的$ G $ - $ actions中的可能增长最小； 2）如果$ g $是一个有限生成的分支组，则必须在空间上实现一个从$ g $中的每个嵌入到强限制类型的同态（例如，有界的自动机组）中的每个同构型； 3）如果$ g $是有限生成的弱分支组，则$ g $不会嵌入间隔交换转换的IET组中。
A subgroup $H$ of a group $G$ is confined if the $G$-orbit of $H$ under conjugation is bounded away from the trivial subgroup in the space $\operatorname{Sub}(G)$ of subgroups of $G$. We prove a commutator lemma for confined subgroups. For groups of homeomorphisms, this provides the exact analogue for confined subgroups (hence in particular for URSs) of the classical commutator lemma for normal subgroups: if $G$ is a group of homeomorphisms of a Hausdorff space $X$ and $H$ is a confined subgroup of $G$, then $H$ contains the derived subgroup of the rigid stabilizer of some open subset of $X$. We apply this commutator lemma in the setting of groups acting on rooted trees. We prove a theorem describing the structure of URSs of weakly branch groups and of their non-topologically free minimal actions. Among the applications of these results, we show: 1) if $G$ is a finitely generated branch group, the $G$-action on $\partial T$ has the smallest possible growth among all faithful $G$-actions; 2) if $G$ is a finitely generated branch group, then every embedding from $G$ into a group of homeomorphisms of strongly bounded type (e.g. a bounded automaton group) must be spatially realized; 3) if $G$ is a finitely generated weakly branch group, then $G$ does not embed into the group IET of interval exchange transformations.
