We say that a group $G$ has Bergman's property (the property of universality of finite width) if for every generating set $X$ of $G$ with $X=X^{-1}$ we have that $G=X^k$ for some natural number $k.$ The property is named after George Bergman who have proved recently that the infinite symmetric groups are groups of universally finite width.
The first example of an infinite group with Bergman's property is due to Shelah (1980s). Lately some other examples have been found: the automorphism groups of doubly transitive chains (Droste-Holland), the automorphism group of reals as a Borel space (Droste-Göbel) and the infinite-dimensional general linear groups over division rings.
In the present paper we prove that the automorphism group $Aut(N)$ of any infinitely generated free nilpotent group $N$ has Bergman's property, is generated by involutions, perfect and has confinality greater than $rank(N).$ Also, we obtain a partial answer to a question posed by Bergman establishing that the automorphism group of a free group of countably infinite rank is a group of universally finite width.
The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on.
Bergman has given the following abstract characterisation of the inner automorphisms of a group $G$: they are exactly those automorphisms of $G$ which can be extended functorially along any homomorphism $G \rightarrow H$ to an automorphism of $H$. This leads naturally to a definition of "inner automorphism" applicable to the objects of any category. Bergman and Hofstra--Parker--Scott have computed these inner automorphisms for various structures including $k$-algebras, monoids, lattices, unital rings, and quandles---showing that, in each case, they are given by an obvious notion of conjugation…
A survey article that presents some recent algebraic and model-theoretic results on the automorphism groups of relatively free groups of infinite rank. The topics include topological aspects, generating sets, descripition of automorpisms and expressive power of the first-order theories.
It has been proven by Schupp and Bergman that the inner automorphisms of groups can be characterized purely categorically as those group automorphisms that can be coherently extended along any outgoing homomorphism. One is thus motivated to define a notion of (categorical) inner automorphism in an arbitrary category, as an automorphism that can be coherently extended along any outgoing morphism, and the theory of such automorphisms forms part of the theory of covariant isotropy. In this paper, we prove that the categorical inner automorphisms in any category $\mathsf{Group}^{\mathcal{J}}$ of p…
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…
Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group.
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.
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.
Let F be an infinitely generated free group and R a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F' of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group Aut(F/R') of the group F/R' is complete. In particular, the automorphism group of any infinitely generated free solvalbe group of derived length at least two is complete. This extends a result by Dyer and Formanek (1977) on finitely generated groups F_n/R' where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.
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.
Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the {\it universality of finite width}: given any generating set $X$ of the symmetric group of an infinite set $Ω,$ there is a uniform bound $k \in \N$ such that any permutation $σ\in \text{Sym}(Ω)$ is a product of at most $k$ elements of $X \cup X^{-1},$ or, in other words, $\text{Sym}(Ω)=(X^{\pm 1})^k.$ Bergman also formulated a sort of general conjecture stating that `the automorphism groups of structures that can be put together out of many isomorphic copies of thems…
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）
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…
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…
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…
The automorphism groups of several of Thompson's countable groups of piecewise linear homeomorphisms of the line and circle are computed and it is shown that the outer automorphism groups of these groups are relatively small. These results can be interpreted as stability results for certain structures of PL functions on the circle. Machinery is developed to relate the structures on the circle to corresponding structures on the line.
Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Out(G,\mathcal{F})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ …
In this paper, we consider the group Aut$(\mathbb{Q}, \leq)$ of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut$(\mathbb{Q}, \leq)$ is contained in an $N$-generated subgroup of Aut$(\mathbb{Q}, \leq)$ for some fixed $N\in\mathbb{N}$. We show that the least such $N$ is $2$. Moreover, for every countable subset of Aut$(\mathbb{Q}, \leq)$, we show that every element can be given as a prescribed product of two generators without using their inverses. More p…
We prove that the unitriangular automorphism group of a free group of rank $n$ has a faithful representation by matrices over a field, or in other words, it is a linear group, if and only if $n \leq 3.$ Thus, we have completed a description of relatively free groups with linear the unitriangular automorphism groups. This description was initiated by Erofeev and the author in \cite{Erofeev}, where proper varieties of groups have been considered.
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with …
We prove that the minimally displaced set of a relatively irreducible automorphism of a free splitting, situated in a deformation space, is uniformly locally finite. The minimally displaced set coincides with the train track points for an irreducible automorphism. We develop the theory in a general setting of deformation spaces of free products, having in mind the study of the action of reducible automorphisms of a free group on the simplicial bordification of Outer Space. For instance, a reducible automorphism will have invariant free factors, act on the corresponding stratum of the bordifica…
We demonstrate the quasi-isometry invariance of two important geometric structures for relatively hyperbolic groups: the coned space and the cusped space. As applications, we produce a JSJ-decomposition for relatively hyperbolic groups which is invariant under quasi-isometries and outer automorphisms, as well as a related splitting of the quasi-isometry groups of relatively hyperbolic groups.
We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreover, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group $C$ can be realized as a group of outer automorphisms of a group $N$, where $N$ is a finitely generated group having Kazhdan's property (T) and containing exactly two conjugacy classes.
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 $F_k$.
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…
We prove that stability -- a strong quasiconvexity property -- pulls back under proper actions on proper metric spaces. This result has several applications, including that convex cocompact subgroups of both mapping class groups and outer automorphism groups of free groups are stable. We also characterize stability in relatively hyperbolic groups whose parabolic subgroups have linear divergence.
Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order $2$. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with ve…
Let $X$ be a smooth hypersurface of dimension $n\geq 1$ and degree $d\geq 3$ in the projective space given as the zero set of a homogeneous form $F$. If $(n,d)\neq (1,3), (2,4)$ it is well known that every automorphism of $X$ extends to an automorphism of the projective space, i.e., $\operatorname{Aut}(X)\subseteq \operatorname{PGL}(n+2,\mathbb{C})$. We say that the automorphism group $\operatorname{Aut}(X)$ is $F$-liftable if there exists a subgroup of $\operatorname{GL}(n+2,\mathbb{C})$ projecting isomorphically onto $\operatorname{Aut}(X)$ and leaving $F$ invariant. Our main result in this …
We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental grou…
Consider a group G and a family $\mathcal{A}$ of subgroups of G. We say that vertex finiteness holds for splittings of G over $\mathcal{A}$ if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in $\mathcal{A}$. We show vertex finiteness when G is a toral relatively hyperbolic group and $\mathcal{A}$ is the family of abelian subgroups. We also show vertex finiteness when G is hyperbolic relative to virtually polycyclic subgroups and $\mathcal{A}$ is the family of virtually cyclic subgroups; if moreover G is one-ended, …
We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $ψ$ and $ϕ$ with dual lamination pairs $Λ^\pm_ψ$ and $Λ^\pm_ϕ$ associated to them, and given a free factor system $\mathcal{F}$ with co-edge number $\geq 2$, $ϕ, ψ$ each preserving $\mathcal{F}$, so that the pair $(ϕ, Λ^\pm_ϕ), (ψ, Λ^\pm_ψ)$ is independent relative to $\mathcal{F}$, then there $\exists$ $M\geq 1$, such that for any integer $m,n \geq M$, the group $\langle ϕ^m, ψ^n \rangle$ is a free group of rank 2, all of whose non-trivial elements except perhaps the po…
Let $G$ be a finite transitive group on a set $Ω$, let $α\in Ω$ and let $G_α$ be the stabilizer of the point $α$ in $G$. In this paper, we are interested in the proportion $$\frac{|\{ω\in Ω\mid ω\textrm{ lies in a }G_α\textrm{-orbit of cardinality at most two}\}|}{|Ω|},$$ that is, the proportion of elements of $Ω$ lying in a suborbit of cardinality at most two. We show that, if this proportion is greater than $5/6$, then each element of $Ω$ lies in a suborbit of cardinality at most two and hence $G$ is classified by a result of Bergman and Lenstra. We also classify the permutation groups attai…
Let $S$ be a compact Riemann surface and $G$ a group of conformal automorphisms of $S$ with $S_0 = S/G$. $S$ is a finite regular branched cover of $S_0$. If $U$ denotes the unit disc, let $Γ$ and $Γ_0$ be the Fuchsian groups with $S = U/Γ$ and $S_0 = U/{Γ_0}$. There is a group homomorphism of $Γ_0$ onto $G$ with kernel $Γ$ and this is termed a surface kernel map. Two surface kernel maps are equivalent if they differ by an automorphism of $Γ_0$. In his 1971 paper Harvey showed that when $G$ is a cyclic group, there is a unique simplest representative for this equivalence class. His result has p…
We prove that for any automorphism $ϕ$ of the restricted wreath product $\mathbb{Z}_2 \mathrm{wr} \mathbb{Z}^k$ and $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d}$ the Reidemeister number $R(ϕ)$ is infinite, i.e. these groups have the property $R_\infty$. For $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d+1}$ and $\mathbb{Z}_p \mathrm{wr} \mathbb{Z}^k$, where $p>3$ is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property $R_\infty$. For these groups and $\mathbb{Z}_m \mathrm{wr} \mathbb{Z}$, where $m$ is relatively prime to $6$, we prove the t…
We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of $p$-groups of lower $p$-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.
We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if…
We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficien…
We show the existence of several new infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten's example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.
Let $F$ be a free group of positive, finite rank and let $Φ\in Aut(F)$ be a polynomial-growth automorphism. Then $F\rtimes_Φ\mathbb Z$ is strongly thick of order $η$, where $η$ is the rate of polynomial growth of $ϕ$. This fact is implicit in work of Macura, but her work predates the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth and thickness explicit. Our result combines with a result independently due to Dahmani-Li, Gautero-Lustig, and Ghosh to show that free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgr…
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually connected real Lie groups, have no nontrivial subgroups with property (T), have finite outer automorphism groups, satisfy Mostow-type Rigidity, have finite asymptotic dimension and rapid decay property, and satisfy Baum-Connes conjecture. We also characterize those lattices in r…
We show that for any non--elementary hyperbolic group $H$ and any finitely presented group $Q$, there exists a short exact sequence $1\to N\to G\to Q\to 1$, where $G$ is a hyperbolic group and $N$ is a quotient group of $H$. As an application we construct a hyperbolic group that has the same $n$--dimensional complex representations as a given finitely generated group, show that adding relations of the form $x^n=1$ to a presentation of a hyperbolic group may drastically change the group even in case $n>> 1$, and prove that some properties (e.g. properties (T) and FA) are not recursively recogni…
