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 is 1.
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 $n\geqslant 3$.
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…
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 work continues the study of the properties of finitely constrained groups of binary tree automorphisms in terms of their Hausdorff dimension. We prove that there are exactly $2^{2d-3}$ finitely constrained groups of binary tree automorphisms with pattern size $d$ and having Hausdorff dimension $1 - \frac{2}{2^{d-1}}$. As part of this proof, we describe the finite patterns that can define such groups, which leads to the fact that all finitely constrained groups of nearly maximal Hausdorff dimension have additive portraits. Additionally, we give an upper bound, in terms of the pattern size …
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.
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 show that the Hausdorff dimension of the closure of the second Grigorchuk group is 43/128. Furthermore we establish that the second Grigorchuk group is super strongly fractal and that its automorphism group equals its normaliser in the full automorphism group of the tree.
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.
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.
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.
If G is a GGS-group defined over a p-adic tree, where p is an odd prime, we calculate the order of the congruence quotients $G_n=G/\Stab_G(n)$ for every n. If G is defined by the vector $e=(e_1,...,e_{p-1})\in\F_p^{p-1}$, the determination of the order of $G_n$ is split into three cases, according as e is non-symmetric, non-constant symmetric, or constant. The formulas that we obtain only depend on p, n, and the rank of the circulant matrix whose first row is e. As a consequence of these formulas, we also obtain the Hausdorff dimension of the closures of all GGS-groups over the p-adic tree.
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…
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 …
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…
In 2003 Bieri and Geoghegan generalized the Bieri-Neuman-Strebel invariant $Σ^1$ by defining $Σ^1(ρ)$, $ρ$ an isometric action by a finitely generated group $G$ on a proper CAT(0) space $M$. In this paper, we show how the natural and well-known connection between Bass-Serre theory and covering space theory provides a framework for the calculation of $Σ^1(ρ)$ when $ρ$ is a cocompact action by $G = B \rtimes A$, $A$ a finitely generated group, on a locally finite Bass-Serre tree $T$ for $A$. This framework leads to a theorem providing conditions for including an endpoint in, or excluding an endp…
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 that a finitely generated pro-$p$ group $G$ acting on a pro-$p$ tree $T$ splits as a free amalgamated pro-$p$ product or a pro-$p$ HNN-extension over an edge stabilizer. If $G$ acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-$p$ group of a finite graph of pro-$p$ groups $(\cal G, Γ)$ with edge and vertex groups being stabilizers of certain vertices and edges of $T$ respectively. If edge stabilizers are procyclic, we give a bound on $Γ$ in terms of the minimal number of generators of $G$. We also give a criterion for a pro-$p$ group $…
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.
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.
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…
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.
We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the regular tree $T_{d}$ of degree $d\in\mathbb{N}_{\ge 3}$. Three applications are given: First, we characterize the automorphism types which the quasi-center of a non-discrete subgroup of $\mathrm{Aut}(T_{d})$ may feature in terms of the group's local~action. In doing so, we explicitly construct closed, non-discrete, compactly generated subgroups of $\mathrm{Au…
Given a locally finite leafless tree $ T $, various algebraic groups over local fields might appear as closed subgroups of $ \text{Aut} (T)$. We show that the set of closed cocompact subgroups of $ \text{Aut} (T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $ \text{Aut} (T)$. This is done via a study of the integral Bruhat$-$Tits model of $ \text{SL}_2 $ and $ \text{SU}_3^{L/K} $, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show that in residue characteristic…
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,…
A group is called $Λ$-free if it has a free Lyndon length function in an ordered abelian group $Λ$, which is equivalent to having a free isometric action on a $Λ$-tree. A group has a regular free length function in $Λ$ if and only if it has a free isometric action on a $Λ$-tree so that all branch points belong to the orbit of the base point. In this paper we prove that every finitely presented $Λ$-free group $G$ can be embedded into a finitely presented group with a regular free length function in $Λ$ so that the length function on $G$ is preserved by the embedding. Next, we prove that every f…
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.
Serre in "Trees" laid down the fundamentals of the theory of groups acting on simplicial trees. In particular, Bass-Serre theory makes it possible to extract information about the structure of a group from its action on a simplicial tree. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat--Tits buildings are trees. In this survey we will discuss the following generalizations of ideas from "Trees": the theory of isometric group actions on $Λ$-trees and the theory of lattices in the product of trees where we describe in more detail results on ari…
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.
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.
