AI 中文总结
该研究通过公理化重叠函数构造自同构对称化的可容许重叠函数，得到局部有限树及$G$的弱分支作用，还描述了$G$的有限弱分支扩张，证明了相关自同构群同构关系
AI 中文摘要
设有限生成群$G$在局部有限根树$T$上有弱分支作用，其边界为$\boldsymbol{\nabla} T$。根树结构由重叠函数编码，该函数是我们对边界上格罗莫夫积的命名：$c(\boldsymbol{\nabla},\boldsymbol{\nabla})=|\boldsymbol{\nabla}\boldsymbol{\nabla}|$。我们对该函数进行公理化，并证明当它“可容许”时，可恢复根树。根据拉夫列纽克和涅克拉舍维奇的边界刚性定理，$\boldsymbol{\nabla}(G)$在$\boldsymbol{\nabla} T$上有典范作用。因此，我们构造重叠函数的自同构对称化：$\boldsymbol{\nabla}c(\boldsymbol{\nabla},\boldsymbol{\nabla})=\boldsymbol{\nabla}_{\boldsymbol{\nabla}(G)}c(\boldsymbol{\nabla}\boldsymbol{\nabla},\boldsymbol{\nabla}\boldsymbol{\nabla})$。我们证明$\boldsymbol{\nabla}c$仍是可容许的$G$-不变重叠函数，且其关联树$\boldsymbol{\nabla} T$是局部有限的。$G$在$\boldsymbol{\nabla} T$上的作用是忠实且弱分支的，当且仅当原$T$上的作用是分支的。此外，$N_{\boldsymbol{\nabla}(\boldsymbol{\nabla} T)}(G)\boldsymbol{\nabla}\boldsymbol{\nabla}(G)$，特别地，$\boldsymbol{\nabla}(G)$是弱分支的。我们还描述了$G$的有限弱分支扩张：它们恰好是$\boldsymbol{\nabla}(G)$的有限子群的拉回。若$G$是分支的，所有这些扩张都是分支的，两种情况下它们都作用于同一棵树$\boldsymbol{\nabla} T$。
英文摘要
Let a finitely generated group $G$ act weakly branch on a locally finite rooted tree $T$ with boundary $\partial T$. The rooted tree structure is encoded by the \textit{overlap function}, which is our name for the Gromov product on the boundary: \[ c(ξ,η)=|ξ\wedgeη|. \] We axiomatise this function and show that when it is `admissible', one can recover the rooted tree. By the boundary rigidity theorem of Lavreniuk and Nekrashevych, $\operatorname{Aut}(G)$ acts canonically on $\partial T$. We therefore form the automorphism symmetrisation of the overlap function: \[ \widehat c(ξ,η) =\inf_{α\in\operatorname{Aut}(G)}c(α\cdotξ,α\cdotη). \] We prove that $ \widehat c $ is again an admissible $G$-invariant overlap function and that its associated tree $ \widehat T $ is locally finite. The action of $G$ on $ \widehat T $ is faithful and weakly branch, and is branch if and only if the original action on $T$ is branch. Moreover, \[ N_{\operatorname{Aut}(\widehat T)}(G) \cong \operatorname{Aut}(G). \] In particular, $ \operatorname{Aut}(G) $ is weakly branch. We also describe the finite weakly branch extensions of $ G $: they are precisely the pullbacks of finite subgroups of $ \operatorname{Out}(G) $. If $ G $ is branch, all these extensions are branch. In both cases, they act on the same tree $ \widehat T $.
Comments19 pages. Comments welcome
