On the subexponential growth of groups acting on rooted trees

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。

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.

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…

Quasi-invariant measures for some amenable groups acting on the line

In this note we show that if $G$ is a solvable group acting on the line, and if there is $T\in G$ having no fixed points, then there is a Radon measure $μ$ on the line quasi-invariant under $G$. In fact, our method allows for the same conclusion for $G$ inside a class of groups that is closed under extensions and contains all solvable groups and all groups of subexponential growth.

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。

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。

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。

Hausdorff dimension of some groups acting on the binary tree

Based on the work of Abercrombie, Barnea and Shalev gave an explicit formula for the Hausdorff dimension of a group acting on a rooted tree. We focus here on the binary tree T. Abert and Virag showed that there exist finitely generated (but not necessarily level-transitive) subgroups of AutT of arbitrary dimension in [0,1]. In this article we explicitly compute the Hausdorff dimension of the level-transitive spinal groups. We then show examples of 3-generated spinal groups which have transcendental Hausdroff dimension, and exhibit a construction of 2-generated groups whose Hausdorff dimension …

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.

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.

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring $\mathbb Z_d$ of $d$-adic integers

We show that every polynomial in $\mathbb Z[x]$ defines an endomorphism of the $d$-ary rooted tree induced by its action on the ring $\mathbb Z_d$ of $d$-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in $\mathbb Z[x]$ of the same degree. In the case of permutational polynomials acting on $\mathbb Z_d$ by bijections the induced endomorphisms are automorphisms of the tree. In the case of $\mathbb Z_2$ such polynomials were completely characterized by Rivest. As our main application we utilize the result of Rivest to derive the condition on the coeffi…

Branch Groups

This is a long introduction to the theory of "branch groups": groups acting on rooted trees which exhibit some self-similarity features in their lattice of subgroups.

Amenability of groups acting on trees

This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation \[Δ= < b,t|[b,t^2]b^{-1},[[[b,t^{-1}],b],b]>.\] I show that it acts transitively on a 3-regular tree, and that $Γ=< b,b^{t^{-1}}$ stabilizes a vertex and acts by restriction on a binary rooted tree. $Γ$ is a "fractal group", generated by a 3-state automaton, and is a discrete analogue of the monodromy action of iterates of f(z)=z^2-1 on associated coverings of the…

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。

Folner sets of alternate directed groups

An explicit family of Folner sets is constructed for some directed groups acting on a rooted tree of sublogarithmic valency by alternate permutations. In the case of bounded valency, these groups were known to be amenable by probabilistic methods. The present construction provides a new and independent proof of amenability, using neither random walks, nor word length.

On a group associated to $z^2-1$

We construct a group acting on a binary rooted tree; this discrete group mimics the monodromy action of iterates of $f(z)=z^2-1$ on associated coverings of the Riemann sphere. We then derive some algebraic properties of the group, and describe for that specific example the connection between group theory, geometry and dynamics. The most striking is probably that the quotient Cayley graphs of the group (aka ``Schreier graphs'') converge to the Julia set of $f$.

Groups of intermediate growth

I describe a class of groups acting on rooted trees. The original claim was that all have intermediate word growth between polynomial and exponential. The argument constructs a functional equation on the growth formal power series, and derives the growth properties from its analytical behaviour. Unfortunately, there are substantial mistakes in the proofs, so the main claims should be considered unsettled. In particular, it is still an open problem whether the growth of the Gupta-Sidki group is intermediate or exponential.

On growth of homology torsion in amenable groups

Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $\tilde X$ with compact quotient $X$. Fix $n \geq 1$, assume $H_n(\tilde X, \mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the integral homology in dimension $n$ of $\tilde{X}/H_i$ grows subexponentially in $[G:H_i]$. By way of contrast, if $X$ is not compact, there are solvable groups of derived length 3 for which torsion in homology can grow faster than any given function.

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…

Multi-EGS-groups：同余商的指数

Multi-EGS-groups: exponent of congruence quotients

本文研究了作用于p进根树的multi-EGS-group，推导出计算所有同余商指数的公式，并证明该公式仅依赖于素数p、层数n及群的周期性。

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…

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…

Groups with near exponential residual finiteness growth

A function $\mathbb{N} \to \mathbb{N}$ is near exponential if it is bounded above and below by functions of the form $2^{n^c}$ for some $c > 0$. In this article we develop tools to recognize the near exponential residual finiteness growth in groups acting on rooted trees. In particular, we show the near exponential residual finiteness growth for certain branch groups, including the first Grigorchuk group, the family of Gupta-Sidki groups and their variations, and Fabrykowski-Gupta groups. We also show that the family of Gupta-Sidki p-groups, for $p\geq 5$, have super-exponential residual finit…

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。

Extensions of amenable groups by recurrent groupoids

We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This covers amenability of a wide class groups, the amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk's group, Basilica group, the full topological group of Cantor minimal system, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, groups that naturally appear in holomorph…

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。

On the finitely generated Hausdorff spectrum of spinal groups

We study the finitely generated Hausdorff spectrum of spinal automorphism groups acting on rooted trees. Given any $α\in [0,1]$, we construct a branch group $G_α$ such that $G_α$ has a finitely generated subgroup $H$ where $H$ has Hausdorff dimension $α$ in $G$. Using results by Barnea, Shalev and Klopsch we further deduce that the finitely generated Hausdorff spectrum of this group $G_α$ contains $\mathcal{L}_α\cup ([0, 1] \cap \mathcal{L})$, where $\mathcal{L}$ is a countable subset of $\mathbb{Q}$ and $\mathcal{L}_α$ is a certain set of countably many irrational numbers in the interval $[0,…

可提升自相似群与尺度群

Liftable self-similar groups and scale groups

本文通过将局部域的等距与伸缩群等同于正则树自同构群的子群，引入可提升自相似群并证明其升HNN扩张的闭包构成尺度群，同时给出多个嵌入2-进数域伸缩群的具体例子。

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。

可达性与孪生宽度

Accessibility and Twin-width

研究有限生成群中有限孪生宽度与可达性的关系，通过结合威尔克斯构造及博内等人的结果，证明存在有限生成群$\Gamma$，其有有限一致孪生宽度但不可达，相应凯莱图也如此。