GGS-groups over primary trees: Branch structures

On maximal subgroups of infinite index in branch and weakly branch groups

We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that every maximal subgroup of the Basilica group is of finite index.

The profinite completion of multi-EGS groups

The class of multi-EGS groups is a generalisation of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally our results show that branch multi-EGS groups are just infinite.

关于分支群有限生成子群的结构

On the structure of finitely generated subgroups of branch groups

本文描述了具有子群归纳性质的分支群（如第一Grigorchuk群和挠GGS群）有限生成子群的块结构。

Portrait growth in contracting, regular branch groups

We address a question of Grigorchuk by providing both a system of recursive formulas and an asymptotic result for the portrait growth of the first Grigorchuk group. The results are obtained through analysis of some features of the branching subgroup structure of the group. More generally, we provide recursive formulas for the portrait growth of any finitely generated, contracting, regular branch group, based on the coset decomposition of the groups that are higher in the branching subgroup structure in terms of the lower subgroups. Using the same general approach we fully describe the portrait…

Engel elements in weakly branch groups

We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all $p$-elements and that the group is virtually a $p$-group (and so periodic) for some prime $p$. We also show that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition. Also, we apply these resu…

分支群的弱极大子群

Weakly maximal subgroups of branch groups

本文研究分支群的弱极大子群，描述了两个族（射线稳定化子的推广及块结构族），并证明对Grigorchuk群和挠性GGS群所有弱极大子群均属此二族，给出多种等价刻画。

分支群的子群归纳性质

Subgroup induction property for branch groups

本文研究了分支群的子群归纳性质，证明了具有该性质的有限生成分支群必为挠群、刚好无限且子群可分，并证明所有挠GGS群均具有此性质，提供了首个无限族例子。

On the Basilica Operation

Inspired by the Basilica group $\mathcal B$, we describe a general construction which allows us to associate to any group of automorphisms $G \leq \operatorname{Aut}(T)$ of a rooted tree $T$ a family of Basilica groups $\operatorname{Bas}_s(G), s \in \mathbb{N}_+$. For the dyadic odometer $\mathcal{O}_2$, one has $\mathcal B = \operatorname{Bas}_2(\mathcal{O}_2)$. We study which properties of groups acting on rooted trees are preserved under this operation. Introducing some techniques for handling $\operatorname{Bas}_s(G)$, in case $G$ fulfills some branching conditions, we are able to calcula…

根树自同构群的pro-$\mathcal{C}$同余性质

Pro-$\mathcal{C}$ congruence properties for groups of rooted tree automorphisms

本文推广了作用于根树的群的同余子群问题，引入了基于有限群伪簇的pro-$\mathcal{C}$完备概念，给出了弱正则分支群满足$\mathcal{C}$-CSP的充要条件，并据此证明了Basilica群及常数定义向量GGS-groups具有$p$-CSP。