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 products. Furthermore, we describe fully the towers of G and H.
More precisely, the tower of G is infinite countable, and the terms of the tower are 2-groups. Quotients of successive terms are infinite elementary abelian 2-groups.
In contrast, the tower of H has length 2, and its terms are {2,3}-groups. We show that the quotient aut^2(H)/aut(H) is an elementary abelian 3-group of countably infinite rank, while aut^3(H)=aut^2(H).
In this thesis I study the automorphism tower of free nilpotent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the automorphism group of the Lie group. This allows me to lift the description of the derivation tower of the free nilpotent Lie algebra to obtain information about the automorphism tower of the free nilpotent group. The main result in this thesis states that the automorphism tower of the free nilpotent group $Γ(n,d)$ on $n$ …
We study properties of eigenvalues of a matrix associated with a randomly chosen partial automorphism of a regular rooted tree. We show that asymptotically, as the numbers of levels goes to infinity, the fraction of non-zero eigenvalues converges to zero in probability.
We prove that the automorphism group of an arbitrary non-abelian free group is complete. It generalizes the result by J.Dyer and E.Formanek (1975) stating the completeness of automorphism group of finitely generated free groups. Using the description of involutions in automorphism groups of free groups (J. Dyer, P. Scott, 1975) we obtain a group-theoretic characterization of inner automorphisms determined by primitive elements in the automorphism group of any non-abelian free group F. It follows that the subgroup Inn(F) is characteristic in Aut(F), and hence the latter one is complete.
We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for all odd $n\ge1003$ and $m>1$. It follows that the group of automorphisms $Aut(B(m,n))$ of the free Burnside group $B(m,n)$ is complete for odd $n\ge1003$, that is it has a trivial center and any automorphism of $Aut(B(m,n))$ is inner. Thus, for groups $B(m,n)$ is solved the automorphism tower problem and is showed that it is as short as the automorphism tower…
We study asymptotics of the spectral measure of a randomly chosen partial automorphism of a rooted tree. To every partial automorphism $x$ we assign its action matrix $A_x$. It is shown that the uniform distribution on eigenvalues of $A_x$ converges weakly in probability to $δ_0$ as $n \to \infty$, where $δ_0$ is the delta measure concentrated at $0$.
The aim of this article is to discuss and clarify the notion of fractality for subgroups of the group of automorphisms of a regular rooted tree. For this purpose we define three types of fractality. We show that they are not equivalent, by giving explicit examples. Furthermore we present some tools that are helpful in order to determine the fractality of a given group.
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.
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…
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group, and so on, iterating transfinitely. Each group maps canonically into the next using inner automorphisms, and so at limit stages one can take a direct limit and continue the iteration. The tower is said to terminate if a fixed point is reached, that is, if a group is reached which is isomorphic to its automorphism group by the natural map. This occurs if a complete group is reached, one which is centerless and has only inner automorphisms. Wielandt [1939] proved the classical…
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…
The automorphism tower of a group is obtained by computing its automorphism group, the automorphism group of THAT group and so on, iterating transfinitely. Each group maps into the next using inner automorphisms and one takes a direct limit at limit stages. The question is whether the process ever terminates in a fixed point, a group which is isomorphic to its automorphism group by the natural map. In this talk I will prove that every group has a terminating automorphism tower. After this, I will discuss the set-theoretic aspects of the height of the automorphism tower of a group, and sketch m…
We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.
It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in this paper, we will show that it is impossible to find an explicit better bound using ZFC.
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.
For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined, the height is the ordinal where this becomes fixed, arriving to a complete group. We first show that for any kappa=kappa^{aleph_0} there is a group of cardinality kappa with height>kappa^+ (improving the lower bound). Second we show that for many such kappa there is a group of height > 2^kappa, so proving that the upper bound essentially cannot be improved.
We show that for any full and sufficiently transitive (i.e. \textit{flexible}) group $G$ of homeomorphisms of Cantor space, $\mathrm{Aut}(\mathrm{Aut}(G)) = \mathrm{Aut}(G)$. This class contains many generalisations of the Higman-Thompson groups $G_{n,r}$, and the Rational group $\mathcal{R}_{2}$ of Grigorchuk, Nekrashevych, and Suchanski{\u ı}. We also demonstrate that for generalisations $T_{n,r}$ of R. Thompson's group $T$, $\mathrm{Aut}(\mathrm{Aut}(T_{n,r}))= \mathrm{Aut}(T_{n,r})$. In the case of the groups $G_{n,r}$ and $T_{n,r}$ our results extend results of Brin and Guzm{\' a}n for Th…
Assume that a finite group $G$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ and complement $H$ such that $C_{G}(F)=1$. In this paper, we investigate this situation and prove that if $C_G(H)$ is supersoluble and $C_{G'}(H)$ is nilpotent, then $G$ is supersoluble. Also, we show that $G$ is a Sylow tower group of a certain type if $C_{G}(H)$ is a Sylow tower group of the same type.
For a centerless group G, we can define its automorphism tower. We define G^{alpha} : G^0=G, G^{alpha +1}=Aut(G^alpha) and for limit ordinals G^delta=bigcup_{alpha < delta}G^alpha . Let tau_G be the ordinal when the sequence stabilizes. Thomas' celebrated theorem says tau_G<2^{|G|})^{+} and more. If we consider Thomas' proof too set theoretical, we have here a shorter proof with little set theory. However, set theoretically we get a parallel theorem without the axiom of choice. We attach to every element in G^alpha, the alpha-th member of the automorphism tower of G, a unique quantifier free t…
Baumslag conjectured in the 1970s that the automorphism tower of a finitely generated free nilpotent group must be very short. Let F_{n,c} denote a free nilpotent group of finite rank n at least two and of nilpotency class c at least two. In 1976 Dyer and Formanek proved that the automorphism group of F_{n,2} is even complete (and hence the height of the aumorphism tower of F_{n,2} is two) provided that n is not three; in the case when n=3, the height of the automorphism tower of F_{n,2} is three. The author proved in 2001 that the automorphism group of any infinitely generated free nilpotent …
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…
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.
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.
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.
Finite dimensional representations of the Hanoi Towers group are used to calculate the spectra of the finite graphs associated to the Hanoi Towers Game on three pegs (the group serves as a renorm group for the game). These graphs are Schreier graphs of the action of the Hanoi Towers group on the levels of the rooted ternary tree. The spectrum of the limiting graph (Schreier graph of the action on the boundary of the tree) is also provided.
It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in the quotient, $H/H(1)$, of $H$ by its uniscalar subgroup has a unique minimal generating set which determines a natural Cayley graph structure on $P$. For each compact, open subgroup $U$ of $G$, a graph is defined and it is shown that if $P$ is multiplicative over $U$ then this graph is a regular, rooted, strongly simple $P$-graph. This extends to higher rank…
We characterize those countable rooted trees whose full automorphism group has uncountable strong cofinality or contains an open subgroup with ample generics.
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…
In this work we study automorphisms of synchronous self-similar groups, the existence of extensions to automorphisms of the full group of automorphisms of the infinite rooted tree on which these groups act on. When they do exist, we obtain conditions for the continuity of such extensions with respect to the depth metric, but we also construct examples of groups where such extensions do not exist. We study the case of the lamplighter group $\mathcal{L}_k = \mathbb{Z}_k \wr \mathbb{Z}$
The structure of a certain subgroup $S$ of the automorphism group of a partially commutative group (RAAG) $G$ is described in detail: namely the subgroup generated by inversions and elementary transvections. We define admissible subsets of the generators of $G$, and show that $S$ is the subgroup of automorphisms which fix all subgroups $\langle Y\rangle$ of $G$, for all admissible subsets $Y$. A decomposition of $S$ as an iterated tower of semi-direct products in given and the structure of the factors of this decomposition described. The construction allows a presentation of $S$ to be computed…
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.
The groups of order 64p without a normal sylow p-subgroup are listed, and their automorphism groups are also determined. As a by-product of our original effort to get these groups, we needed to determine the automorphism groups of those groups of order 64 with an odd-order automorphism. In view of the fact that we already had determined these groups and that these automorphism groups are not given explicitly in the literature, we have appended to this report these automorphism groups. In another project we were looking for new complete groups by following automorphism group towers up to comple…
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.
Let $p\ge 3$ be a prime. A generalised multi-edge spinal group is a subgroup of the automorphism group of a regular $p$-adic rooted tree T that is generated by one rooted automorphism and $p$ families of directed automorphisms, each family sharing a common directed path disjoint from the paths of the other families. This notion generalises the concepts of multi-edge spinal groups, including the widely studied GGS-groups, and the extended Gupta-Sidki groups that were introduced by Pervova. Extending techniques that were developed in these more special cases, we prove: generalised multi-edge spi…
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…
We introduce the $k$-peg Hanoi automorphisms and Hanoi self-similar groups, a generalization of the Hanoi Towers groups, and give conditions for them to be contractive. We analyze the limit spaces of a particular family of contracting Hanoi groups, $H_c^{(k)}$, and show that these are the unique maximal contracting Hanoi groups under a suitable symmetry condition. Finally, we provide partial results on the contraction of Hanoi groups with weaker symmetry.
The group A_{m} of automophisms of a one-rooted m-ary tree admits a diagonal monomorphism which we denote by x. Let A be an abelian state-closed (or self-similar) subgroup of A_{m}. We prove that the combined diagonal and tree-topological closure A* of A is additively a finitely presented Z_m [[x]]-module where Z_m is the ring of m-adic integers. Moreover, if A* is torsion-free then it is a finitely generated pro-m group. The group A splits over its torsion subgroup. We study in detail the case where A* corresponds to a cyclic Z_m[[x]]-module and when m is a prime number, we show A* to be conj…
Università di Padova（帕多瓦大学）；University of the Basque Country UPV/EHU（巴斯克地区大学）；Heinrich Heine University Düsseldorf（杜塞尔多夫海因里希·海涅大学）；Università di Salerno（萨莱诺大学）
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…
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…
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…
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.
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…
Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed point group G^X to lower bounds for the growth in G. We give applications to S-arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.
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 prove that the group of almost-automorphisms of the infinite rooted regular $d$-ary tree $\mathcal{T}_d$ arises naturally as the Thompson-like group of a so called $d$-ary cloning system. A similar phenomenon occurs for any Röver-Nekrashevych group $V_d(G)$, for $G\le Aut(\mathcal{T}_d)$ a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for $G$, is of type $F_\infty$. Namely, we find some natural conditions on subgroups of $G$ to ensure that $V_d(G)$ is of type $F_\infty$, and in particular we prove…
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.
The term "strong approximation" is used to describe phenomena where an arithmetic group as well as all of its Zariski dense subgroups have a large image in the congruence quotients. We exhibit analogues of such phenomena in a probabilistic, rather than arithmetic, setting. Let T be the binary rooted tree, Aut(T) its automorphism group. To a given m-tuple a = {a_1,a_2,...,a_m} in Aut(T), we associate a tower of 2m-regular Schreier graphs ...X_n-->X_{n-1}-->...-->X_0. The vertices of X_n are the n^{th} level of the tree and two such are connected by an edge if a generator takes one to the other.…
For each prime p and a monic polynomial f, invertible over p, we define a group G_{p,f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group G_{2,x^2+x+1}. We show that the constructed groups are self-similar, regular branch groups. This enables us to calculate the Hausdorff dimension of their closures, providing concrete examples (not using random methods) of topologically finitely generated closed subgroups of the group of p-adic automorphisms with Hausdorff dimension arbitrarily close to 1. We provide a…
Let $φ\colonΓ\to G$ be a homomorphism of groups. We consider factorizations $Γ\xrightarrow{f} M\xrightarrow{g} G$ of $φ$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have…
We compute the complexity of two infinite families of finite graphs: the Sierpiński graphs, which are finite approximations of the well-known Sierpińsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.
