Branch Groups

Free subgroups in groups acting on rooted trees

We show that if a group $G$ acting faithfully on a rooted tree $T$ has a free subgroup, then either there exists a point $w$ of the boundary $\partial T$ and a free subgroup of $G$ with trivial stabilizer of $w$, or there exists $w\in\partial T$ and a free subgroup of $G$ fixing $w$ and acting faithfully on arbitrarily small neighborhoods of $w$. This can be used to prove absence of free subgroups for different known classes of groups. For instance, we prove that iterated monodromy groups of expanding coverings have no free subgroups and give another proof of a theorem by S. Sidki.

Ergodic decomposition of group actions on rooted trees

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. A special attention is given to the case of groups generated by finite automata. Few examples, including the lamplighter group, Sushchansky group, and the, …

群作用于根树上的虚拟 retract

Virtual retracts in groups acting on rooted trees

本文研究群作用于根树上的虚拟 retract，证明有限生成分支群不具有 LR 性质，并揭示 LR 性质在后临界有限二次复多项式中与欧几里得 orbifold 的关系，同时提供新的 pro-2 群例子。

The Automorphism Tower of Groups acting on Rooted Trees

The group of isometries W of a regular rooted tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in W. This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group G studied by R. Grigorchuk, and the group H studied by N. Gupta and the second author. In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as G and H. We describe this tower for all groups acting on the binary rooted tree which decompose as infinitely iterated wreath product…

On quasi-2-transitive actions of branch groups

An action of a group $G$ on a set $X$ is said to be quasi-n-transitive if the diagonal action of $G$ on $X^n$ has only finitely many orbits. We show that branch groups, a special class of groups of automorphisms of rooted trees, cannot act quasi-2-transitively on infinite sets.

Affine Automorphisms of Rooted Trees

We introduce a class of automorphisms of rooted $d$-regular trees arising from affine actions on their boundaries viewed as infinite dimensional vector spaces. This class includes, in particular, many examples of self-similar realizations of lamplighter groups. We show that for a regular binary tree this class coincides with the normalizer of the group of all spherically homogeneous automorphisms of this tree: automorphisms whose states coincide at all vertices of each level. We study in detail a nontrivial example of an automaton group that contains an index two subgroup with elements from th…

受限子群的交换子引理及其在作用于根树上的群中的应用

A commutator lemma for confined subgroups and applications to groups acting on rooted trees

本文证明受限子群的交换子引理，并据此刻画弱分支群的 URS 结构，进而证明有限生成分支群作用增长最小、嵌入强有界型同胚群必空间实现以及不嵌入 IET 群等结论。

通过重叠函数实现弱分支作用的自同构不变精细化

Automorphism-invariant refinements of weakly branch actions via overlap functions

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

Quasi-automorphisms of the infinite rooted 2-edge-coloured binary tree

We study the group $QV$, the self-maps of the infinite $2$-edge coloured binary tree which preserve the edge and colour relations at cofinitely many locations. We introduce related groups $QF$, $QT$, $\tilde{Q}T$, and $\tilde{Q}V$, prove that $QF$, $\tilde{Q}T$, and $\tilde{Q}V$ are of type $\mathrm{F}_\infty$, and calculate finite presentations for them. We calculate the normal subgroup structure and rational homology of all $5$ groups, the Bieri--Neumann--Strebel--Renz invariants of $QF$, and discuss the relationship of all $5$ groups with other generalisations of Thompson's groups.

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。

Coxeter groups, symmetries, and rooted representations

Let $(W,S)$ be a Coxeter system, let $G$ be a group of symmetries of $(W,S)$ and let $f : W \to \GL (V)$ be the linear representation associated with a root basis $(V, \langle .,. \rangle, Π)$.We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $Π$ and $\langle .,. \rangle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde Π\subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde Π)$ is a root basis of $W^G$, and we show that the induced representation $f^G : W^G \to \GL(V^G)$ is the linear representation associated with $(V^G, \langle .,. \rangle, \tilde Π)$.I…

Non-abelian free groups admit non-essentially free actions on rooted trees

We show that every countable non-abelian free group $Γ$ admits a spherically transitive action on a rooted tree $T$ such that the action of $Γ$ on the boundary of $T$ is not essentially free. This reproves a result of Bergeron and Gaboriau. The existence of such an action answers a question of Grigorchuk, Nekrashevich and Sushchanskii.

L-presentations and branch groups

We introduce L-presentations: group presentations given by a generating set, a set of relations and a set of substitution rules on the generating set producing more relations. We first study in full generality the structure of finitely L-presented groups, i.e. groups for which all the above sets are finite, and then show that a broad class of groups acting on rooted trees admit explicitly constructible finite L-presentations: they are the "branch" groups defined by R. Grigorchuk.

Modeling rooted in-trees by finite p-groups

The aim of this chapter is to provide an adequate graph theoretic framework for the description of periodic bifurcations which have recently been discovered in descendant trees of finite p-groups. The graph theoretic concepts of rooted in-trees with weighted vertices and edges perfectly admit an abstract formulation of the group theoretic notions of successive extensions, nuclear rank, multifurcation, and step size. Since all graphs in this chapter are infinite and dense, we use methods of pattern recognition and independent component analysis to reduce the complex structure to periodically re…

Dimension and randomness in groups acting on rooted trees

We explore the structure of the p-adic automorphism group Gamma of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turan. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups of Gamma generated by three random elements are full-dimensional and that there exist finitely generated subgroups of arbitrary di…

生成自由群的自相似群的动力学与阻碍

Freeness Obstructions for Self-Similar Groups Acting on Rooted Trees

该研究证明自相似群自由性的两个一般阻碍，推导自动机群相关结论，明确自由性与双可逆性的关联，回答Grigorchuk问题并扩展Sidki的相关结果。

Maximal subgroups of multi-edge spinal groups

A multi-edge spinal group is a subgroup of the automorphism group of a regular p-adic rooted tree, generated by one rooted automorphism and a finite number of directed automorphisms sharing a common directing path. We prove that torsion multi-edge spinal groups do not have maximal subgroups of infinite index. This generalizes a result of Pervova for GGS-groups.

On the conjugacy problem for finite-state automorphisms of regular rooted trees

We study the conjugacy problem in the automorphism group $Aut(T)$ of a regular rooted tree $T$ and in its subgroup $FAut(T)$ of finite-state automorphisms. We show that under the contracting condition and the finiteness of what we call the orbit-signalizer, two finite-state automorphisms are conjugate in $Aut(T)$ if and only if they are conjugate in $FAut(T)$, and that this problem is decidable. We prove that both these conditions are satisfied by bounded automorphisms and establish that the (simultaneous) conjugacy problem in the group of bounded automata is decidable.

GGS-groups over primary trees: Branch structures

We study branch structures in Grigorchuk-Gupta-Sidki groups (GGS-groups) over primary trees, that is, regular rooted trees of degree $p^n$ for a prime $p$. Apart from a small set of exceptions for $p=2$, we prove that all these groups are weakly regular branch over $G''$. Furthermore, in most cases they are actually regular branch over $γ_3(G)$. This is a significant extension of previously known results regarding periodic GGS-groups over primary trees and general GGS-groups in the case $n=1$. We also show that, as in the case $n=1$, a GGS-group generated by a constant vector is not branch.

Trees, unsplittability, property (FA) and the likes

The paper is devoted to a study of certain fixed point properties, and their relatives, in the context of full automorphism groups of countable rooted trees. Namely, we study Serre's property (FA'), also called unsplittability, property (FA), the uncountable strong cofinality, and ample generics. We give a new proof of a theorem of Psaltis to the extent that automorphism groups of rooted trees are unsplittable, and show under what circumstances automorphism groups of rooted trees have uncountable strong cofinality (and thus property (FA).) Also, we prove a necessary and sufficient condition fo…

The Liouville property for groups acting on rooted trees

We show that on groups generated by bounded activity automata, every symmetric, finitely supported probability measure has the Liouville property. More generally we show this for every group of automorphisms of bounded type of a rooted tree. For automaton groups, we also give a uniform upper bound for the entropy of convolutions of every symmetric, finitely supported measure.

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…

关于作用于根树的群的有限生成豪斯多夫谱

On finitely generated Hausdorff spectra of groups acting on rooted trees

该研究证明有限生成正则分支群闭包的有限生成豪斯多夫谱为[0,1]，回答了Klopsch和Shalev的问题，推广了Abért-Virág结果，首次给出指定维数的显式子群，并定义良滤过解决相关问题。

The congruence subgroup problem for branch groups

We state and study the congruence subgroup problem for groups acting on rooted tree, and for branch groups in particular. The problem is reduced to the computation of the congruence kernel, which we split into two parts: the branch kernel and the rigid kernel. In the case of regular branch groups, we prove that the first one is Abelian while the second has finite exponent. We also establish some rigidity results concerning these kernels. We work out explicitly known and new examples of non-trivial congruence kernels, describing in each case the group action. The Hanoi tower group receives part…

Two periodicity conditions for spinal groups

A constant spinal group is a subgroup of the automorphism group of a regular rooted tree, generated by a group of rooted automorphisms $A$ and a group of directed automorphisms $B$ whose action on a subtree is equal to the global action. We provide two conditions in terms of certain dynamical systems determined by $A$ and $B$ for constant spinal groups to be periodic, generalising previous results on Grigorchuk--Gupta--Sidki groups and other related constructions. This allows us to provide various new examples of finitely generated infinite periodic groups.

Groups acting on rooted trees and their representations on the boundary

We consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree. We introduce and discuss the new notion of locally 2-transitive actions. Assuming local 2-transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents. The method can be used to study Gelfand pairs and enables us to answer a question of Grigorchuk. To provide examples, we analyse in detail the local 2-transitivity of GGS-groups. Moreover, our res…

Rooted trees, strong cofinality and ample generics

We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.

Weakly maximal subgroups in regular branch groups

Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups tha…

Orbit length generating functions of automorphisms of a rooted regular binary tree

To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function Φ_w that encodes information on the orbit structure of w. We prove that this is a rational function if w can be described by finitely many recursion relations of a particular form. We show that this condition is satisfied for all elements of the discrete iterated monodromy group Γassociated to a postcritically finite quadratic polynomial over C. For such Γwe also prove that there are only finitely many possibilities for the denominator of Φ_w, and we describe a procedure to determi…

Maximal subgroups of non-torsion Grigorchuk-Gupta-Sidki groups

A Grigorchuk-Gupta-Sidki (GGS-)group is a subgroup of the automorphism group of the $p$-adic tree for an odd prime $p$, generated by one rooted automorphism and one directed automorphism. Pervova proved that all torsion GGS-groups do not have maximal subgroups of infinite index. Here we extend the result to non-torsion GGS-groups, which include the weakly regular branch, but not branch, GGS-group.

弱分支群的非闭子群

Non-closed subgroups of weakly branch groups

针对作用于足够正则根树的弱分支群G，构造两类非闭子群连续族：一类含非ERF子群，另一类在额外假设下含ERF子群，均在G的完备或同余拓扑中不闭。

分支作用与结构格

Branch actions and the structure lattice

本文研究了分支群的结构格与树边界闭开子集布尔代数之间的同构关系，推广了Wilson关于第二类群的结论。

Finitely constrained groups of maximal Hausdorff dimension

We prove that if G_P is a finitely constrained group of binary rooted tree automorphisms (a group binary tree subshift of finite type) defined by an essential pattern group P of pattern size d, d>1, and if G_P has maximal Hausdorff dimension (equal to 1-1/2^{d-1}), then G_P is not topologically finitely generated. We describe precisely all essential pattern groups P that yield finitely constrained groups with maximal Haudorff dimension. For a given size d, d>1, there are exactly 2^{d-1} such pattern groups and they are all maximal in the group of automorphisms of the finite rooted regular tree…

On diagonal actions of branch groups and the corresponding characters

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the associated rooted tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II$_1$-factor representations, and infinitely many continuous ergodic invariant random subgroups.

关于有限生成的Engel分支群

On finitely generated Engel branch groups

本文构造了有限生成的Engel分支群，回答了相关存在性问题，并提供了继Golod之后第二类有限生成非幂零Engel群的例子，方法基于根树作用与词长快速收缩。

The automorphism group of a multi-GGS group

A multi-GGS-group is a group of automorphisms of a regular rooted tree, generalising the Gupta--Sidki $p$-groups. We compute the automorphism groups of all non-constant multi-GGS-groups.

Some Distributions on Finite Rooted Binary Trees

We introduce some natural families of distributions on rooted binary ranked plane trees with a view toward unifying ideas from various fields, including macroevolution, epidemiology, computational group theory, search algorithms and other fields. In the process we introduce the notions of split-exchangeability and plane-invariance of a general Markov splitting model in order to readily obtain probabilities over various equivalence classes of trees that arise in statistics, phylogenetics, epidemiology and group theory.

The Spectral Problem, Substitutions and Iterated Monodromy

We provide a self-similar measure for the self-similar group $G$ acting faithfully on the binary rooted tree, defined as the iterated monodromy group of the quadratic polynomial $z^2+i$. We also provide an $L$-presentation for $G$ and calculations related to the spectrum of the Markov operator on the Schreier graph of the action of $G$ on the orbit of a point on the boundary of the binary rooted tree.

On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3.

Automata generating free products of groups of order 2

We construct a family of automata with n states, n>3, acting on a rooted binary tree that generate the free products of cyclic groups of order 2.

一类迭代单值群的极大子群

Maximal subgroups of a family of iterated monodromy groups

针对迭代单值群中与广义Basilica群相似的子群族，结合分支群到弱分支群的极大子群研究进展，证明该子群族的极大子群仅具有限指数。

Profinite iterated monodromy groups arising from quadratic morphisms with infinite postcritical orbits

We study in detail the profinite group G arising as geometric étale iterated monodromy group of an arbitrary quadratic morphism f with an infinite postcritical orbit over a field of characteristic different from two. This is a self-similar closed subgroup of the group of automorphisms of a regular rooted binary tree. In many cases it is equal to the automorphism group of the tree, but there remain some interesting cases where it is not. In these cases we prove that the conjugacy class of G depends only on the combinatorial type of the postcritical orbit of f. We also determine the Hausdorff di…

收缩分形群、Schreier图与谱

A Contracting Fractal Group, Schreier Graphs and The Spectra

本文研究了一个由5状态自动机生成的收缩分形群，分析了其自相似作用的特征，展示了对应的Schreier图，并数值近似了极限图上离散拉普拉斯算子的谱。

尺度群

Scale Groups

本文证明了正则树等距群中固定端点且顶点传递的闭子群，与根树上的自复制群及具有正尺度的全不连通局部紧群元素之间存在对应关系。

一类非收缩弱正则分支群中的分支群

A Branch Group in a Class of Non-Contracting Weakly Regular Branch Groups

本文构造了一类包含无限族分形弱正则分支群的非收缩群，并研究其阿贝尔化、恰好无限性及字问题，首次给出显式非收缩分支群实例，证明其指数增长。

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.

Minimal generating sets in wreath products

We find some sufficient conditions under which the permutational wreath product of two groups has a minimal generating set. In particular we prove that for a regular rooted tree the group of all automorphisms and the group of all finite-state automorphisms of such a tree satisfy these conditions.

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。

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…

On amenability of automata groups

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability just of a certain explicit family of groups ("Mother groups") which is done by analyzing the asymptotic properties of random walks on these groups.

Schreier graphs of the Basilica group

With any self-similar action of a finitely generated group $G$ of automorphisms of a regular rooted tree $T$ can be naturally associated an infinite sequence of finite graphs $\{Γ_n\}_{n\geq 1}$, where $Γ_n$ is the Schreier graph of the action of $G$ on the $n$-th level of $T$. Moreover, the action of $G$ on $\partial T$ gives rise to orbital Schreier graphs $Γ_ξ$, $ξ\in \partial T$. Denoting by $ξ_n$ the prefix of length $n$ of the infinite ray $ξ$, the rooted graph $(Γ_ξ,ξ)$ is then the limit of the sequence of finite rooted graphs $\{(Γ_n,ξ_n)\}_{n\geq 1}$ in the sense of pointed Gromov-Hau…

仅它们的基数可阻止群成为弱分支群的商群

Only their cardinality can prevent groups from being quotients of weakly branch groups

该研究证明基数不超过$2^{\aleph_0}$的群均为弱分支群的商群，给出生成元数量相关结论，并应用解决了Thillaisundaram等人的两个群论问题。

Homomorphic images of Branch groups, and Serre's property (FA)

It is shown that a finitely generated branch group has Serre's property (FA) if and only if it does not surject onto the infinite cyclic group or the infinite dihedral group. An example of a finitely generated self-similar branch group surjecting onto the infinite cyclic group is constructed.

On self-similarity of $p$-adic analytic pro-$p$ groups of small dimension

Given a torsion-free $p$-adic analytic pro-$p$ group $G$ with $\mathrm{dim}(G) < p$, we show that the self-similar actions of $G$ on regular rooted trees can be studied through the virtual endomorphisms of the associated $\mathbb{Z}_p$-Lie lattice. We explicitly classify 3-dimensional unsolvable $\mathbb{Z}_p$-Lie lattices for $p$ odd, and study their virtual endomorphisms. Together with Lazard's correspondence, this allows us to classify 3-dimensional unsolvable torsion-free $p$-adic analytic pro-$p$ groups for $p\geqslant 5$, and to determine which of them admit a faithful self-similar actio…