Using the canonical JSJ splitting, we describe the outer automorphism group $\Out(G)$ of a one-ended word hyperbolic group $G$. In particular, we discuss to what extent $\Out(G)$ is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups $\Out(G)$ is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in $\Out(G)$, for $G$ any torsion-free hyperbolic group.
More generally, let $Γ$ be a finite graph of groups decomposition of an arbitrary group $G$ such that edge groups $G_e$ are rigid (i.e\. $\Out(G_e)$ is finite). We describe the group of automorphisms of $G$ preserving $Γ$, by comparing it to direct products of suitably defined mapping class groups of vertex groups.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.
We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular $\mathrm{Aut}(\mathbb{F}_n)$ is acylindrically hyperbolic for every $n\ge 2$. More generally, if $G$ is a group which is not virtually cyclic, and hyperbolic relative to a finite collection $\mathcal{P}$ of finitely generated proper subgroups, then $\mathrm{Aut}(G,\mathcal{P})$ is acylindrically hyperbolic. As a consequence, a free-by-cyclic group $\mathbb{F}_n\rtimes_φ\mathbb{Z}$ is acylindrically hyperbolic if and only if $φ$ has infinite order in $\mathrm{Out}(…
We prove that the group STame($k^3$) of special tame automorphisms of the affine 3-space is not simple, over any base field of characteristic zero. Our proof is based on the study of the geometry of a 2-dimensional simply-connected simplicial complex C on which the tame automorphism group acts naturally. We prove that C is contractible and Gromov-hyperbolic, and we prove that Tame($k^3$) is acylindrically hyperbolic by finding explicit loxodromic weakly proper discontinuous elements.
An automorphism $α$ of a group $G$ is normal if it fixes every normal subgroup of $G$ setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we prove that for any relatively hyperbolic group $G$, $Inn(G)$ has finite index in the subgroup $Aut_n(G)$ of normal automorphisms. If, in addition, $G$ is non-elementary and has no non-trivial finite normal subgroups, then $Aut_n(G)=Inn(G)$. As an application, we show that $Out(G)$ is residually finite for every finitely generated residually finite group $G$ with more than one end.
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We also generalize the main result for relatively hyperbolic groups.
We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually p-residually finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely-ended and virtually p-residually finite.
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…
In this article we study automorphisms and endomorphisms of lacunary hyperbolic groups. We prove that every lacunary hyperbolic group is Hopfian, answering a question by Henry Wilton. In addition, we show that if a lacunary hyperbolic group has the fix point property for actions on $\mathbf R$-trees, then it is co-Hopfian and its outer automorphism group is locally finite. We also construct lacunary hyperbolic groups whose automorphism group is infinite, locally finite, and contains any locally finite group given in advance.
We prove that the mapping torus group $\FN \rtimes_α \Z$ of any automorphism $α$ of a free group $\FN$ of finite rank $n \geq 2$ is weakly hyperbolic relative to the canonical (up to conjugation) family $\mathcal H(α)$ of subgroups of $\FN$ which consists of (and contains representatives of all) conjugacy classes that grow polynomially under iteration of $α$. Furthermore, we show that $\FN \rtimes_α \Z$ is strongly hyperbolic relative to the mapping torus of the family $\mathcal H(α)$. As an application, we use a result of Drutu-Sapir to deduce that $\FN \rtimes_α \Z$ has Rapic Decay.
We prove that an automorphism $ϕ:F\to F$ of a finitely generated free group $F$ is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes.
We show that word hyperbolicity of automorphism groups of graph products $G_Γ$ and of Coxeter groups $W_Γ$ depends strongly on the shape of the defining graph $Γ$. We also characterized those $Aut(G_Γ)$ and $Aut(W_Γ)$ in terms of $Γ$ that are virtually free.
We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced HNN-extensions which are not residually finite. We prove that this method can never yield a "new" counter-example to Gromov's conjecture on the residual finiteness of hyperbolic groups.
A connected Lie group admitting an expansive automorphism is known to be nilpotent, but all nilpotent Lie groups do not admit expansive automorphism. In this article, we find sufficient conditions for a class of nilpotent Lie groups to admit expansive automorphism.
Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follows that the fixed subgroup is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. If all peripheral subgroups of G are slender (respectively, slender and coherent), our result implies that the fixed subgroup of every automorphism of G is finitely generated (respectively, finitely presented). In particular, this hap…
We are concerned with orderable groups and particularly those with orderings invariant not only under multiplication, but also under a given automorphism or family of automorphisms. Several applications to topology are given: we prove that the fundamental groups of hyperbolic nonorientable surfaces, and the groups of certain fibred knots are bi-orderable. Moreover, we show that the pure braid groups associated with hyperbolic nonorientable surfaces are left-orderable.
We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.
Based on the concept of dual cones introduced by J.Opgenorth we give an algorithm to compute a generating system of the group of automorphisms of a lattice endowed with a hyperbolic bilinear form.
Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture
Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay（巴黎萨克雷大学）；Université de Caen Normandie, CNRS, Laboratoire de Mathématiques Nicolas Oresme（诺曼底大学）；Rheinische Friedrich-Wilhelms-Universität Bonn（波恩大学）；University of Oxford（牛津大学）；Univ. Lyon, ENS de Lyon, UMPA UMR 5669（里昂大学）
In this article, we show that some negative curvature may survive when taking the automorphism group of a finitely generated group. More precisely, we prove that the automorphism group $\mathrm{Aut}(G)$ of a one-ended hyperbolic group $G$ turns out to be acylindrically hyperbolic. As a consequence, given a group $H$ and a morphism $φ: H \to \mathrm{Aut}(G)$, we deduce that the semidirect product $G \rtimes_φH$ is acylindrically hyperbolic if and only if $\mathrm{ker}(H \oversetφ{\to} \mathrm{Aut}(G) \to \mathrm{Out}(G))$ is finite.
We prove that for a free product $G$ with free factor system $\mathcal{G}$, any automorphism $ϕ$ preserving $\mathcal{G}$, atoroidal (in a sense relative to $\mathcal{G}$) and none of whose power send two different conjugates of subgroups in $\mathcal{G}$ on conjugates of themselves by the same element, gives rise to a semidirect product $G\rtimes_ϕ\mathbb{Z}$ that is relatively hyperbolic with respect to suspensions of groups in $\mathcal{G}$. We recover a theorem of Gautero-Lustig and Ghosh that, if $G$ is a free group, $ϕ$ an automorphism of $G$, and $\mathcal{G}$ is its family of polynomia…
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 $3$…
Part 1 : We remark that the conjugacy problem for pairs of hyperbolic au- tomorphisms of a finitely presented group (typically a free group) is decidable. The solution that we propose uses the isomorphism problem for the suspensions, and the study of their automorphism group. Part 2 : In a previous work (part 1), we remarked that the conjugacy problem for pairs of atoroidal automorphisms of a free group was solvable by mean of the isomorphism problem for hyperbolic groups and an orbit problem for the automorphism group of their suspensions (i.e. their semidirect product with Z for the relevant…
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathcal X$ is proper, we get a compactification of $\mathcal X$; we also prove that our construction generalizes the Gromov boundary of a hyperbolic space. In our first main set of applications, we introduce a notion of geometrical finiteness for hierarchically hyperbolic subgroups of…
We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that…
On K3 surfaces with hyperbolic automorphism groups
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；KUIAS and Research Institute for Mathematical Sciences, Kyoto University（京都大学宇宙地球生命研究所及数学科学研究所）；Mathematical Sciences, the University of Tokyo（东京大学理学部数学科）；National Center for Theoretical Sciences, Mathematics Division, National Taiwan University（台湾大学理论科学中心数学组）；Center for Applied Mathematics and KL-AAGDM, Tianjin University（天津大学应用数学中心与代数几何动力学重点实验室）
研究Picard数≥6的复射影K3表面的Néron-Severi晶格有限性，基于 genus 一纤维和Kikuta与Takatsu近期工作，证明其有限性。
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 …
For any finite collection $f_i$ of fully irreducible automorphisms of the free group $F_n$ we construct a connected $δ$-hyperbolic $Out(F_n)$-complex in which each $f_i$ has positive translation length.
We show that, if $H$ is a random subgroup of a finitely generated free group $F_k$, only inner automorphisms of $F_k$ may leave $H$ invariant. A similar result holds for random subgroups of toral relatively hyperbolic groups, more generally of groups which are hyperbolic relative to slender subgroups. These results follow from non-existence of splittings over slender groups which are relative to a random group element. Random subgroups are defined using random walks or balls in a Cayley tree of $F_k$.
This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if $Γ$ is a finite graph which contains at least two vertices and is not a join and if $\mathcal{G}$ is a collection of finitely generated irreducible groups, then either $Γ\mathcal{G}$ is infinite dihedral or $\mathrm{Aut}(Γ\mathcal{G})$ is acylindrically hyperbolic. This theorem is new even for right-angled Artin groups and right-angled Coxeter groups. Various consequences are deduced from this statement and from the techniques used to prove it…
Let $ϕ$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_ϕ = F_n \rtimes_ϕ \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite graph $K_{3,3}$ into $\partial G_ϕ$. To do so, we define the directional Whitehead graph and prove that an indecomposable $F_n$-tree is Levitt type if and only if one of its directional Whitehead graphs contains more than one edge. As an application, we obtain a direct proof of Kapovich-Kleiner's theorem that $\partial G_ϕ$ is homeomorphic to the Menger curve if the auto…
It is well-known that there is a faithful representation of braid groups on automorphism groups of free groups, and it is also well-known that free groups are bi-orderable. We investigate which n-strand braids give rise to automorphisms which preserve some bi-ordering of the free group rank n. As a consequence of our work we find that of the two minimal volume hyperbolic 2-cusped orientable 3-manifolds, one has bi-orderable fundamental group whereas the other does not. We prove a similar result for the 1-cusped case, and have further results for more cusps. In addition, we study pseudo-Anosov …
We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G.
We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non…
We prove that the group of outer automorphisms of the free Coxeter group $W_n$ is acylindrically hyperbolic in the sense of Osin. As an application, we observe that any CAT(0) space admitting a geometric action by Out($W_n$) must contain a rank-one geodesic. The theorem proceeds from expanding on a well-known relationship between Out($W_n$) and the outer automorphism group of free groups.
We prove that numerous negatively curved simply connected locally compact polyhedral complexes, admitting a discrete cocompact group of automorphisms, have automorphism groups which are locally compact, uncountable, non linear and virtually simple. Examples include hyperbolic buildings, Cayley graphs of word hyperbolic Coxeter systems, and generalizations of cubical complexes, that we call even polyhedral complexes. We use tools introduced by Tits in the case of automorphism groups of trees, and Davis-Moussong's geometric realisation of Coxeter systems.
We find a set of generators for the automorphism group of a graph product of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup of star-automorphisms defined in [7]. We follow closely the plan of M. Laurence's paper [11].
The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct w…
We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic loca…
The outer automorphism group Out(G) of a group G acts on the set of conjugacy classes of elements of G. McCool proved that the stabilizer $Mc(c_1,...,c_n)$ of a finite set of conjugacy classes is finitely presented when G is free. More generally, we consider the group $Mc(H_1,...,H_n)$ of outer automorphisms $Φ$ of G acting trivially on a family of subgroups $H_i$, in the sense that $Φ$ has representatives $α_i$ with $α_i$ equal to the identity on $H_i$. When G is a toral relatively hyperbolic group, we show that these two definitions lead to the same subgroups of Out(G), which we call "McCool…
Similarly to the action of $Out(F_N)$ on Outer Space, the outer automorphism group of a Generalized Baumslag Solitar group acts on a deformation space endowed with the Lipschitz metric and the action of any fully irreducible automorphism with a train track representative is hyperbolic. Inspired by previous work by Algom-Kfir, we prove that the axis of such an automorphism is strongly contracting.
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 construct a nonuniform lattice and an infinite family of uniform lattices in the automorphism group of a hyperbolic building with all links a fixed finite building of rank 2 associated to a Chevalley group. We use complexes of groups and basic facts about spherical buildings.
A hyperbolic lattice is called \textit{$1.2$-reflective} if the subgroup of its automorphism group generated by all $1$- and $2$-reflections is of finite index. The main result of this article is a complete classification of $1.2$-reflective maximal anisotropic lattices of rank $4$.
