Automorphism groups of locally finite trees provide a large class of examples of simple totally disconnected locally compact groups. It is desirable to understand the connections between the global and local structure of such a group. Topologically, the local structure is given by the commensurability class of a vertex stabiliser; on the other hand, the action on the tree suggests that the local structure should correspond to the local action of a stabiliser of a vertex on its neighbours.
We study the interplay between these different aspects for the special class of groups satisfying Tits' independence property. We show that such a group has few open subgroups if and only if it acts locally primitively. Moreover, we show that it always admits many germs of automorphisms which do not extend to automorphisms, from which we deduce a negative answer to a question by George Willis. Finally, under suitable assumptions, we compute the full group of germs of automorphisms; in some specific cases, these turn out to be simple and compactly generated, thereby providing a new infinite family of examples which generalise Neretin's group of spheromorphisms. Our methods describe more generally the abstract commensurator group for a large family of self-replicating profinite branch groups.
We generalize a result of T. Koberda by showing that the natural action of the automorphism group on the space of left-orderings is faithful for all nonabelian bi-orderable groups G, as well as for a certain class of left-orderable groups that includes the braid groups. As a corollary we show that the action of the automorphism group of G on the boundary of G is faithful whenever G is bi-orderable and hyperbolic. We also analyze the action of the commensurator of G on its space of virtual left-orderings.
For each Sophie Germain prime $g \geq 5,$ we construct an absolutely simple polarized abelian variety of dimension $g$ over a finite field, whose automorphism group is a cyclic group of order $4g+2$. We also provide a description on the asymptotic behavior of prime numbers that are related to our construction.
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 construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new exampl…
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…
Motivated by the theory of Riemann surfaces, we classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed points on each non-regular orbit and at most four fixed points in total. In each case we also give information about the branching datum of the surface. There is a shorter version of this article (submitted for publication), and in this extended version we give many more details about the GAP code that we use for the calculations. We also explicitly include a le…
Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with $R(q) < G \leq \mathsf{Aut}(R(q))$ is not a C-group and therefore not the automorphism group of an abstract regular polytope of any rank.
We study centralisers of finite order automorphisms of generalisations of Thompson's group F and conjugacy classes of finite subgroups in finite extensions of these groups. In particular, we show that centralisers of finite automorphisms in these generalised Thompson groups either admit a finite type classifying space or are not finitely generated. As an application we deduce results about the Bredon type of such finite extensions.
We study groups of bimeromorphic and biholomorphic automorphisms of projective hyperkähler manifolds. Using an action of these groups on some non-positively curved space, we deduce many of their properties, including finite presentation, strong form of Tits' alternative and some structural results about groups consisting of transformations with infinite order.
We introduce the notion of über-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of non-simplicity, of groups acting on non locally compact spaces of arbitrary dimension. We also give a simple local criterion to construct über-contracting elements for groups acting on complexes with unbounded links of vertices. As an application, we show the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$, a subgroup of the $3$…
We study the group Tame(SL$_2$) of tame automorphisms of a smooth affine 3-dimensional quadric, which we can view as the underlying variety of SL(2,$\mathbb{C}$). We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT(0) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame(SL$_2$) is linearizable, and that Tame(SL$_2$) satisfies the Tits alternative.
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…
Groups preserving a distributive product are encountered often in algebra. Examples include automorphism groups of associative and nonassociative rings, classical groups, and automorphism groups of p-groups. While the great variety of such products precludes any realistic hope of describing the general structure of the groups that preserve them, it is reasonable to expect that insight may be gained from an examination of the universal distributive products: tensor products. We give a detailed description of the groups preserving tensor products over semisimple and semiprimary rings, and presen…
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 introduce the notion of the $k$-closure of a group of automorphisms of a locally finite tree, and give several examples of the construction. We show that the $k$-closure satisfies a new property of automorphism groups of trees that generalises Tits' Property $P$. We prove that, apart from some degenerate cases, any non-discrete group acting on a tree with this property contains an abstractly simple subgroup.
We complete the study of finite and profinite groups admitting an action by an elementary abelian group under which the centralizers of automorphisms consist of Engel elements. In particular, we prove the following theorems. Let $q$ be a prime and $A$ an elementary abelian $q$-group of order at least $q^2$ acting coprimely on a profinite group $G$. Assume that all elements in $C_{G}(a)$ are Engel in $G$ for each $a\in A^{\#}$. Then $G$ is locally nilpotent (Theorem B2). Let $q$ be a prime, $n$ a positive integer and $A$ an elementary abelian group of order $q^3$ acting coprimely on a finite gr…
Let $FH$ be a Frobenius group with kernel $F$ and complement $H$, acting coprimely on the finite solvable group $G$ by automorphisms. We prove that if $C_{G}(H)$ is of Fitting length $n$ then the index of the $n$-th Fitting subgroup $F_{n}(G)$ in $G$ is bounded in terms of $|C_{G}(F)|$ and $|F|.$ This generalizes a result of Khukhro and Makarenko \cite{k-m} which handles the case $n=1.$
We study contraction groups for automorphisms of totally disconnected locally compcat groups using the scale of the automorphism as a tool. The contraction group is shown to be unbounded when the inverse automorphism has non-trivial scale and this scale is shown to be the inverse value of the modular function on the closure of the contraction group at the automorphism. The closure of the contraction group is represented as acting on a homogenous tree and closed contraction groups are characterised.
A scale-multiplicative semigroup in a totally disconnected, locally compact group $G$ is one for which the restriction of the scale function on $G$ is multiplicative. The maximal scale-multiplicative semigroups in groups acting 2-transitively on the set of ends of trees without leaves are determined in this paper and shown to correspond to geometric features of the tree.
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…
We describe a group theoretic condition which ensures that any cellular action of a group satisfying this condition on a CAT(0) cube complex has a global fixed point. In particular, we show that this fixed point criterion is satisfied by the automorphism group of a free group of rank n. For the unique subgroup of index two in the automorphism group of a free group, we obtain a similar result.
We introduce a group naturally acting on aperiodic necklaces of length $n$ with two colours using the 1--1 correspondences between aperiodic necklaces and irreducible polynomials over the field $\F_2$ of two elements. We notice that this group is isomorphic to the quotient group of non-degenerate circulant matrices of size $n$ over that field modulo a natural cyclic subgroup. Our groups turn out to be isomorphic to the sandpile groups for a special sequence of directed graphs.
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.
In this paper, we generalize the tools that were introduced in [Dar19b] in order to study the Andreadakis problem for subgroups of IAn. In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IAn. We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group on n-1 generators , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.
It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M\_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M\_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E\_i\times E\_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,9…
We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting minimally on them. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group.
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.
A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the tetravalent vertex-transitive non-Cayley graphs of order $6p$ are classified for each prime $p$.
Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Property (T), is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.
Let S be a compact Riemann surfaces of genus g >= 2 and G a conformal automoprhism group of order n acting on S. In this paper we give the definition of an adapted generating set and an adapted basis for the first homology group of such a compact Riemann surface. This generating set and basis reflect the action of G in as simple manner as possible. This can be seen in the matrix of the action of G which we obtain. We prove the existence of such a generating set and basis for any conformal group acting on such a surface and find the matrix. This extends our earlier results on adapted bases and …
This work is a continuation of Automorphisms of $K$-groups I, P. Flavell, preprint. The main object of study is a finite $K$-group $G$ that admits an elementary abelian group $A$ acting coprimely. For certain group theoretic properties $\mathcal P$, we study the $AC_{G}(A)$-invariant $\mathcal P$-subgroups of $G$. A number of results of McBride, 'Near solvable signalizer functors on finite groups' J. Algebra {\bf 78}(1) (1982) 181-214 and 'Nonsolvable signalizer functors on finite groups', J. Algebra {\bf 78}(1) (1982) 215-238 are extended. One purpose of this work is to build a general theory…
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 …
Let $q$ be a prime and $A$ an elementary abelian $q$-group acting as a coprime group of automorphisms on a profinite group $G$. We show that if $A$ is of order $q^2$ and some power of each element in $C_G(a)$ is Engel in $G$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Assuming that $A$ is of order $q^3$ we prove that if some power of each element in $C_G(a)$ is Engel in $C_G(a)$ for any $a\in A^{\#}$, then $G$ is locally virtually nilpotent. Some analogues consequences of quantitative nature for finite groups are also obtained.
We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n^{1/2+ε} for the minimum size of the edges in such a hypergraph. This is essentially best possible.
A group with a geometric action on some hyperbolic space is necessarily word hyperbolic, but on the other hand every countable group acts (metrically) properly by isometries on a locally finite hyperbolic graph. In this paper we consider what happens when a group acts isometrically on a restricted class of hyperbolic spaces, for instance quasitrees. We obtain strong conclusions on the group structure if the action has a locally finite orbit, especially if the group is finitely generated. We also look at group actions on finite products of quasitrees, where our actions may be by automorphisms o…
Let $T$ be a locally finite tree all of whose vertices have valency at least $6$. We classify, up to isomorphism, the closed subgroups of $\mathrm{Aut}(T)$ acting $2$-transitively on the set of ends of $T$ and whose local action at each vertex contains the alternating group. The outcome of the classification for a fixed tree $T$ is a countable family of groups, all containing two remarkable subgroups: a simple subgroup of index $\leq 8$ and (the semiregular analog of) the universal locally alternating group of Burger-Mozes (with possibly infinite index). We also provide an explicit example sho…
Let $A$ be an elementary abelian group of order $p^{k}$ with $k\geq 3$ acting on a finite $p'$-group $G$. The following results are proved. If $γ_{k-2}(C_{G}(a))$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then $γ_{k-2}(G)$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. If, for some integer $d$ such that $2^{d}+2\leq k$, the $d$th derived group of $C_{G}(a)$ is nilpotent of class at most $c$ for any $a\in A^{#}$, then the $d$th derived group $G^{(d)}$ is nilpotent and has $\{c,k,p\}$-bounded nilpotency class. Earlier this was known only in the case where $k\leq 4$.
We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group $\Z$) and the one-sided case (spatially acting monoid $\N$, alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sof…
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…
In this paper, we show that if G is a finite p-group (p prime) acting by automorphisms on a $δ$-hyperbolic Poincare Duality group, then the fixed subgroup is a Poincare Duality group over Z/p. We also provide examples to show that the fixed subgroup might not even be a Duality group over Z.
Let $N=L_n(q)$, {$n \geq 2$}, $q$ a prime power, be a projective linear simple group. We classify all Steiner quadruple systems admitting a group $G$ with $N \leq G \leq \Aut(N)$. In particular, we show that $G$ cannot act as a group of automorphisms on any Steiner quadruple system for $n>2$.
We characterize those Lie groups, and algebraic groups over a local field of characteristic zero, whose first reduced L^p-cohomology is zero for all p>1, extending a result of Pansu. As an application, we obtain a description of Gromov-hyperbolic groups among those groups. In particular we prove that any non-elementary Gromov-hyperbolic algebraic group over a non-Archimedean local field of zero characteristic is quasi-isometric to a 3-regular tree. We also extend the study to semidirect products of a general locally compact group by a cyclic group acting by contracting automorphisms.
We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL_n(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a h…
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.
This paper begins the classification of all edge-primitive 3-arc-transitive graphs by classifying all such graphs where the automorphism group is an almost simple group with socle an alternating or sporadic group, and all such graphs where the automorphism group is an almost simple classical group with a vertex-stabiliser acting faithfully on the set of neighbours.
This is a study of algebras with involution that become isomorphic over a separable closure of the base field to a tensor product of two composition algebras. We classify these algebras, provide criteria for isomorphism and isotopy, and determine their automorphism groups, structure groups, and norm-similitude groups. The most interesting cases of these algebras are the $64$-dimensional bi-octonion algebras, on which groups of absolute type $(G_2 \times G_2)\rtimes \mathbb{Z}/2\mathbb{Z}$ and $\mathbf{Spin}_{14}$ act faithfully by automorphisms and norm-preserving isotopies, respectively. We c…
We provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees. In particular, we consider amalgamated products and HNN-extensions, 1-relator groups, automorphism groups of polynomial algebras, 3-manifold groups and graph products. Acylindrical hyperbolicity is then used to obtain some results about the algebraic structure, analytic properties and measure equivalence rigidity of groups from these classes.
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 consider the group action of the automorphism group $\I_n=\aut(\Zz_n)$ on the set $\Zz_n$, that is the set of residue classes modulo $n$. Clearly, this group action provides a representation of $\I_n$ as a permutation group acting on $n$ points. One problem to be solved regarding this group action is to find its cycle index. Once it is found, there appears a vast class of related enumerative and computational problems with interesting applications. We provided the cycle index of specified group action in two ways. One of them is more abstract and hence compact, while another one is basicall…
