Pierre Fima, François Le Maître, Soyoung Moon, Yves Stalder
We establish a sharp sufficient condition for groups acting on trees to be highly transitive when the action on the tree is minimal of general type. This gives new examples of highly transitive groups, including icc non-solvable Baumslag-Solitar groups, thus answering a question of Hull and Osin.
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We prove that if the group generated by a Mealy automaton acts level-transitively on a regular rooted tree, then the semigroup generated by the dual automaton has exponential growth, hence giving a decision procedure of exponential growth for a restricted family of automaton semigroups.
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…
In 2008, Borovik and Cherlin posed the problem of showing that the degree of generic transitivity of an infinite permutation group of finite Morley rank $(X,G)$ is at most $n+2$ where $n$ is the Morley rank of $X$. Moreover, they conjectured that the bound is only achieved (assuming transitivity) by $\operatorname{PGL}_{n+1}(\mathbb{F})$ acting naturally on projective $n$-space. We solve the problem under the two additional hypotheses that (1) $(X,G)$ is $2$-transitive, and (2) $(X-\{x\},G_x)$ has a definable quotient equivalent to $(\mathbb{P}^{n-1}(\mathbb{F}),\operatorname{PGL}_{n}(\mathbb{…
Boundary dynamics, triple transitivity, and mixed identities in weakly hyperbolic groups
研究在δ-双曲空间上具有一般类型作用且诱导作用忠实于Gromov边界极限集的群，根据拓扑自由性划分亚类，证明拓扑自由等价于无混合恒等式，并应用于Kac-Moody群。
Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building $Δ$ and K be the stabilizer of a special vertex in $Δ$. It is known that (G, K) is a Gelfand pair as soon as G acts strongly transitively on $Δ$; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse to this statement, namely: if (G, K) is a Gelfand pair and G acts cocompactly on $Δ$, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in G and their peculia…
It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme …
We study under which condition an amalgamated free product or an HNN-extension over a finite subgroup admits an amenable, transitive and faithful action on an infinite countable set. We show that such an action exists if the initial groups admit an amenable and almost free action with infinite orbits (e.g. virtually free groups or infinite amenable groups). Our result relies on the Baire category Theorem. We extend the result to groups acting on trees.
Let $Ω=\{1,2,...,n\}$ where $n \ge 2$. The {\em shape} of an ordered set partition $P=(P_1,..., P_k)$ of $Ω$ is the integer partition $λ=(λ_1,...,λ_k)$ defined by $λ_i = |P_i|$. Let G be a group of permutations acting on $Ω$. For a fixed partition $λ$ of n, we say that G is {\em $λ$-transitive} if G has only one orbit when acting on partitions P of shape $\la$. A corresponding definition can also be given when G is just a set. For example, if $λ=(n-t,1,...,1)$, then a $λ$-transitive group is the same as a t-transitive permutation group and if $λ=(n-t,t)$, then we recover the t-homogeneous perm…
The transitivity degree of a group $G$ is the supremum of all integers $k$ such that $G$ admits a faithful $k$-transitive action. Few obstructions are known to impose an upper bound on the transitivity degree for infinite groups. The results of this article provide two new classes of groups whose transitivity degree can be computed, as a corollary of a classification of all $3$-transitive actions of these groups. More precisely, suppose that $G$ is a subgroup of the homeomorphism group of the circle $\mathsf{Homeo}(\mathbb{S}^1)$ or the automorphism group of a tree $\mathsf{Aut}(\mathbb{T})$. …
In this paper we study the action of the fundamental group of a finite metric graph on its universal covering tree. We assume the graph is finite, connected and the degree of each vertex is at least three. Further, we assume an irrationality condition on the edge lengths. We obtain an asymptotic for the number of elements in a fixed conjugacy class for which the associated displacement of a given base vertex in the universal covering tree is at most $T$. Under a mild extra assumption we also obtain a polynomial error term.
We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms of the $d$-ary rooted tree $T$. We prove that a nontrivial ergodic IRS of $\mathrm{Alt}_f(T)$ that acts without fixed points on the boundary of $T$ contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regular branch group contains the derived subgroup of a generalized rigid level stabilizer. We also prove that every weakly bra…
We show that every group in a large family of (not necessarily torsion) spinal groups acting on the ternary rooted tree is of subexponential growth.
We present some partial results concerning a-T-menability of groups acting on trees. Various known results are given uniform proofs.
We generalise results of Thomas, Allcock, Thom-Petersen, and Kar-Niblo to the first $\ell^2$-Betti number of quotients of certain groups acting on trees by subgroups with free actions on the edge sets of the graphs.
A finitely generated group $G$ acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag--Solitar group ($GBS$ group). We prove that a 1-knot group $G$ is $GBS$ group iff $G$ is a torus-knot group and describe all n-knot GBS groups for $n\geqslant 3$.
Bounding the degree of generic sharp transitivity
本研究在L-群条件下证明了有限Morley秩泛型锐t-传递置换群的次数满足t≤r+2，推进了Borovik-Cherlin猜想，证明基于三个独立的群作用相关关键要素。
We study cocompact lattices with dense projections in a product $G_1 \times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only finitely many discrete subgroups of $G_1 \times G_2$.
Homomorphic Images of Locally Compact Groups Acting on Trees and Buildings
本文研究全不连通局部紧群，通过推广Cartan分解，证明作用于树和建筑的一大类群的连续同态像均为闭集。
Groups acting freely on Z^n-trees (Z^n-free groups) play a key role in the study of non-archimedean group actions. Following Stallings' ideas, we develop graph-theoretic techniques to investigate subgroup structure of Z^n-free groups. As an immediate application of the presented method, we give an effective solution to the Uniform Membership Problem and the Power Problem in Z^n-free groups.
Let T be a k-regular tree (k>2) and A its automorphism group. We analyze a generic finitely generated subgroup Gamma of A. We show that Gamma is free and establish a trichotomy on the closure of Gamma: it is either discrete, compact or has index at most 2 in A.
We examine the question of which finitely generated groups act properly on a finite product of simplicial trees, considering both arbitrary trees and where all trees are locally finite. In the second case we present evidence in favour of hyperbolic surface groups having such an action. However we also present evidence that many RAAGs do not admit such an action and we give an example of a virtually special group which does not act properly preserving factors on any finite product of locally finite trees, even though it does so on a product of three trees without the local finiteness condition.
We investigate the profinite completions of a certain family of groups acting on trees. It turns out that for some of the groups considered, the completions coincide with the closures of the groups in the full group of tree automorphisms. However, we introduce an infinite series of groups for which that is not so, and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups.
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.
This paper further develops the theory of arbitrary semigroups acting on trees via elliptic mappings. A key tool is the Lyndon-Chiswell length function L for the semigroup S which allows one to construct a tree T and an action of S on T via elliptic maps. Improving on previous results, the length function of the action will also be L.
We prove an acylindrical accessibility theorem for finitely generated groups acting on $\mathbf R$-trees. Namely, we show that if $G$ is a freely indecomposable non-cyclic $k$-generated group acting minimally and $M$-acylindrically on an $\mathbf R$-tree $X$ then for any $ε>0$ there is a finite subtree $Y_ε\subseteq X$ of measure at most $2M(k-1)+ε$ such that $GY_ε=X$. This generalizes theorems of Z.Sela and T.Delzant about actions on simplicial trees.
We prove the following theorem: Let $π$ be the fundamental group of a finite graph of groups with finitely generated vertex groups $G_v$ having asdim $G_v\le n$ for all vertices $v$. Then asdim$π\le n+1$. This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.
We describe a simple criterion for showing that a group has Serre's property FA. By exhibiting a certain pattern of finite subgroups, we show that this criterion is satisfied by Aut(F_n) and SL(n,Z) when n>=3.
We introduce two families of examples of groups acting on trees, one consisting of group amalgamations and the other consisting of HNN-extensions, motivated by the problems of $C^*$-simplicity and unique trace property. Moreover, we prove that our examples are not inner amenable and identify a relatively large, simple, normal subgroup in each one.
Pro-$\mathcal{C}$ groups acting on profinite trees
研究Pro - $\mathcal{C}$子群嵌入自由有限完备积的条件，在适当假设下推广到在有限完备树上$k$ - 无圈作用的有限完备群，且结果可应用于重要有限完备群类。
Let $\mathcal{T}_{q+1}$ be the $(q+1)$-regular tree and let $G$ be a group of automorphisms acting transitively on the vertices and on the boundary of $\mathcal{T}_{q+1}$. We give an upper bound for the growth of cocycles with values in any unitary representation of the group $G$. This bound is optimal by projecting the Haagerup cocycle onto an appropriate subspace of $\ell^{2}(E)$. We also obtain a description of functions conditionally of negative type which are unbounded.
We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generated simple groups of finite asymptotic dimension (actually one) not containing lattices. We also obtain examples of simple groups with simple lattices, and we prove the existence of (infinitely many) finitely generated simple groups of asymptotic dimension one. We also prove various properties of these groups, including the existence of a proper action on a CAT(0) cube complex.
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 …
We construct new examples of groups with cohomological dimension 2 and geometric dimension 3 with respect to the families of finite subgroups, virtually abelian groups of bounded rank, and the family of virtually poly-cyclic subgroups. Our main ingredients are the examples constructed by Brady-Leary-Nucinckis and Fluch-Leary, and Bass-Serre theory.
In our paper arXiv:1310.6289, we stated that acylindrical hyperbolicity of a group is invariant under commensurability up to finite kernels. Unfortunately, the proof of this fact contained a gap. The goal of this erratum is to point out the gap and to correct the statements and proofs of results of arXiv:1310.6289 and the subsequent paper arXiv:1501.03066 affected by it.
We construct the first example of a finitely generated group which has Serre's property (FA) (i.e., whenever it acts on a simplicial tree it fixes a vertex), but admits a fixed point-free action on an $\mathbb{R}$-tree with finite arc stabilizers. We also give a short and elementary construction of finitely generated groups that have property (FA) but do not have (F$\mathbb{R}$).
We present an algorithm to decide whether or not a finitely generated subgroup of the isometry group of a locally finite simplicial tree is both discrete and free. The correctness of this algorithm relies on the following conjecture: every `minimal' $n$-tuple of isometries of a simplicial tree either contains an elliptic element or satisfies the hypotheses of the Ping Pong Lemma. We prove this conjecture for $n=2,3$, and show that it implies a generalisation of Ihara's Theorem.
Radu groups acting on trees are CCR
本文分类了在边界上2-传递作用且局部作用包含交错群的树的闭单自同构群的不可约酉表示，并由此证实了Nebbia关于一类半正则树的CCR猜想。
Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $Δ$ of dimension $\geq 2$ that acts strongly transitively on $Δ$ is Moufang. If moreover $Δ$ is irreducible and $H$ is topologically simple, we show that $H$ is the subgroup $\G(k)^+$ of the $k$-rational points $\G(k)$ of the isotropic simple algebraic group $\G$ over a non-Archimedean local field $k$ associated with $Δ$. Secondly, we generalise the proof given in \cite{BM00b} for the case of bi-regular trees to any locally finite thick affine building $Δ$, and obtain tha…
Let $T$ be a locally finite tree without vertices of degree $1$. We show that among the closed subgroups of $\mathrm{Aut}(T)$ acting with a bounded number of orbits, the Chabauty-closure of the set of topologically simple groups is the set of groups without proper open subgroup of finite index. Moreover, if all vertices of $T$ have degree $\geq 3$, then the set of isomorphism classes of topologically simple closed subgroups of $\mathrm{Aut}(T)$ acting doubly transitively on $\partial T$ carries a natural compact Hausdorff topology inherited from Chabauty. Some of our considerations are valid i…
Let $G$ be a group acting on a tree $T$ with finite edge stabilizers of bounded order. We provide, in some very interesting cases, upper bounds for the complexity of the intersection $H\cap K$ of two tame subgroups $H$ and $K$ of $G$ in terms of the complexities of $H$ and $K$. In particular, we obtain bounds for the Kurosh rank $Kr(H\cap K)$ of the intersection in terms of Kurosh ranks $Kr(H)$ and $Kr(K)$, in the case where $H$ and $K$ act freely on the edges of $T$.
We study the class of groups generated by automata that act essentially freely on the boundary of a rooted tree. In the process we establish and discuss some general tools for determining if a group belongs to this class, and explore the connections of this class to the classes of just-infinite and scale-invariant groups. Our main application is a complete classification of groups generated by 3-state automata over 2-letter alphabet that are in this class.
Let G be a finitely generated infinite pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. Then we prove that G splits as a pro-p amalgamated product or as a pro-p HNN-extension over an edge stabilizer. Using this result we prove under certain conditions that free pro-p products with procyclic amalgamation inherit from its free factors the property of each 2-generated subgroup being free pro-p. This generalizes known pro-p results, as well as some pro-p analogs of classical results in abstract combinatorial group theory.
This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree. A zeta function associated to a non-uniform tree lattice with appropriate Hilbert representation is defined. Zeta functions are defined for infinite graphs with a cocompact or finite covolume group action.
It is shown that for any action of a finitely presented group $G$ on an $\R$-tree, there is a decomposition of $G$ as the fundamental group of a graph of groups related to this action. If the action of $G$ on $T$ is non-trivial, i.e. there is no global fixed point, then $G$ has a non-trivial action on a simplcial $\R $-tree.
We study non-ascending HNN extensions acting on their Bass-Serre trees, and characterize C*-simplicity and the unique trace property by means of the kernel and quasi-kernels of the HNN extension in question. We also present a concrete example of an HNN extension that is a new example of a group that is not C*-simple but does have the unique trace property. Additionally, we include certain more general results, mostly based on previous work of various authors, concerning C*-simplicity of groups admitting extreme boundary actions, and in particular, groups acting on trees.
Let $T$ be a tree and $e$ an edge in $T$. If $C$ is a component of $T\setminus e$ and both $C$ and its complement are infinite we say that $C$ is a half-tree. The main result of this paper is that if $G$ is a closed subgroup of the automorphism group of $T$ and $G$ leaves no non-trivial subtree invariant and fixes no end of $T$ then the subgroup generated by the pointwise stabilizers of half-trees is topologically simple. This result is used to derive analogues of recent results of Caprace and De Medts (2011) and it is also applied in the study of the full automorphism group of a locally finit…
An introduction to the local-to-global behaviour of groups acting on trees and the theory of local action diagrams
本文介绍了群在树上作用的局部到全局行为及局部作用图理论，该理论类似于 Bass--Serre Theory，可完全描述具有 Tits' Independence Property (P) 的树上自同构闭群，并为未来研究提供思路。
On the Boone--Higman Conjecture for groups acting on locally finite trees
本文提出一种证明作用于局部有限树的群的Boone--Higman猜想的新方法，并成功证明了所有Baumslag--Solitar群和自由（有限秩）循环群的猜想，且该方法有望推广至更多群类。
The Scale of ($P$)-closed Groups Acting On Trees
本文通过局部作用图描述($P$)-闭群的平移和轴，确定其尺度函数，并刻画幺模和单尺度($P$)-闭群。
Groups Acting Acylindrically on Trees
本文研究了在树上实现圆柱形且非元素作用的群，分类了图积群和某些双曲 3-流形的基本群，并提供了具有树作用但无非元素圆柱形树作用的圆柱形双曲群的例子。
We propose a new look at the Julg-Valette theorem on K-theoretic amenability for groups operating on trees. The main tool is a generalization of a construction of uniformly bounded representations of the free groups due to Pytlik and Szwarc.
Discrete (P)-closed Groups Acting On Trees
本文基于局部作用图，给出了作用在树上的群为离散的充要条件，该群满足Tits独立性性质(P)。
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。
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with …
We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is actually true for infinite vertex stabilizers and some more general, finite of infinite, edge stabilizers that we call highly core-free. We study the notion of highly core-free subgroups and give some examples. In the case of amalgamated free products over highly core-free subgroups and HNN extensions with highly core-free base groups we obtain a genericity result for faithful and highly transitive …
In this note, we give new examples of type I groups generalizing a previous result of Ol'shanskii. More precisely, we prove that all closed non-compact subgroups of Aut(T_d) acting transitively on the vertices and on the boundary of a d-regular tree and satisfying Tits' independence property are type I groups. We claim no originality as we use standard ingredients: the polar decomposition of those groups and the admissibility of all their irreducible unitary representations.
We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto $\mathbf{Z}$. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a genera…
We consider actions of completely metrisable groups on simplicial trees in the context of the Bass--Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild conditions on a completely metrisable group $G$, namely, that the set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any decomposition as a free product with amalgamation, $G=A\ast_CB$, the amalgamated groups $A$, $B$ and $C$ are open in $G$.
Groups acting on products of locally finite trees
研究有限生成群在有限局部有限 simplicial 树积上的作用，支持超曲面群具有此类作用，并明确嵌入 genus 2 闭超曲面群到 SL_2(F_p(x,y))。
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…
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