Outer automorphism groups of certain 1-relator groups

Grossman first showed that outer automorphism groups of 1-relator groups given by orientable surface groups are residually finite, whence mapping class groups of orientable surfaces are residually finite. Allenby, Kim and Tang showed that outer automorphism groups of cyclically pinched 1-relator groups are residually finite, whence mapping class groups of orientable and non-orientable surfaces are residually finite. In this paper we show that outer automorphism groups of certain conjugacy separable 1-relator groups are residually finite.     

Determining whether a finite group of outer automorphisms of a free group is geometric

In this article we give necessary and sufficient conditions for a given finite group of outer automorphisms to be induced by the action of a group of orientation-preserving homeomorphisms on the fundamental group of a punctured surface. When the group is abelian, necessary and sufficient conditions can also be given in the absence of orientability assumptions. These properties are formulated in terms of the finite automorphism groups which project into the given outer automorphism group: each non-trivial automorphism in any such group can fix at most a cyclic subgroup of the fundamental group.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

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 residually finite.     

Small conjugacy classes in the automorphism groups of relatively free groups

Let F be an infinitely generated free group and let R be 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 of the group F/R^′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] 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.     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002     

On a theorem of Burnside on fixed-point-free automorphisms

We prove that a finite group having a fixed-point-free automorphism in the Fitting subgroup of its automorphism group must be abelian of rather restricted structure. As a consequence, no finite nonabelian group could have a fixed-point-free automorphism in the Frattini subgroup of its automorphism group. Received: 21 April 2007, Revised: 18 May 2007     

Almost fixed-point-free automorphisms of soluble groups

Suppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m=4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].     

Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]ⁱ, a,b,c,d∈G and i?0, then the entire derived subgroup G^′ satisfies a positive law. In fact, G^′ is an extension of a nilpotent group by a locally finite group of finite exponent.     

Collapsing groups and positive laws

The paper concerns the question of A. Shalev: is it true that every collapsing group satisfies a positive law? We give a positive answer for groups in a large class C, including all soluble and residually finite groups.     

Embedding groups of graph automorphisms in surfaces

Necessary and sufficient conditions (n.a.s.c) are found for a subgroup of the automorphism group of a finite graph to be realizable as the restriction to an invariant spine of some group of homeomorphisms of a compact surface. Also, n.a.s.c. are found for the restricted case when the surface is required to be orientable. The conditions are formulated in terms of the action of stabilizers of vertices on their stars. In both cases, a parametrization of the possible representations is given. Several examples are treated, as well as an application to deciding whether a given finite group of outer automorphisms of a free group is realizable via a surface homeomorphism.     

有限秩的可解群的正则自同构

设G是有限秩的剩余有限可解群或是有限秩的剩余有限可解群的有限扩张,α是G的一个索数p阶正则自同构且φ:G→G(g→[g,α])是满射,则G是幂零类不超过h(p)的幂零群,其中h(p)是只与p有关的函数.     

Automorphisms of groups

The survey presents classical assertions due to Nielsen, Whitehead, and others, well-known theorems on automorphisms included in monographs on group theory, and recent results in this area. Attention is focused on the progress in automorphism groups theory for free, solvable, modular, and profinite groups. New tools of investigation using graphs and geometrical ideas are also discussed.     

Commutators in Residually Finite Groups

 The following result is proved. Let G be a residually finite group satisfying the identity ([x ₁, x ₂][x ₃, x ₄]) ⁿ  ≡ 1 for a positive integer n that is not divisible by p ² q ² for any distinct primes p and q. Then G′ is locally finite. Received 7 May 2001; in revised form 3 December 2001     

On groups whose subgroups are closed in the profinite topology

A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

On words that are concise in residually finite groups

A group-word w$w$ is called concise if whenever the set of w$w$-values in a group G$G$ is finite it always follows that the verbal subgroup w(G)$w\left(G\right)$ is finite. More generally, a word w$w$ is said to be concise in a class of groups X$X$ if whenever the set of w$w$-values is finite for a group G∈X$G\in X$, it always follows that w(G)$w\left(G\right)$ is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w$w$ is a multilinear commutator and q$q$ is a prime-power, then the word w^q$w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where w=_γk$w={\gamma }_k$ the word w^q$w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q$w^q$ is actually concise in the class of all groups.     

Automorphisms of groups and of schemes of finite type

We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free. (A group virtually has a property if some subgroup of finite index has it.) The rest of the paper is devoted to a study of the groups of automorphisms. Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ. This is an old idea (cf. the discussion of Magnus in [11]). In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietyS _n (Γ) on which Out(Γ) acts. We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free. This is true, for example, when Γ is a free group, or a surface group. In the latter case Out(Γ) is a “mapping class group.” Partially supported by the NSF under Grant MCS 80-05802.     

Groupes Moyennables, Dimension Topologique Moyenne et sous-décalages

Let G be an infinite countable residually finite amenable group. In this paper we construct a continuous action of G on a compact metrisable space X such that the dynamical system (X, G) cannot be embedded in the G-shift on [0,1] ^G . This result generalizes a construction due to E. Lindenstrauss and B. Weiss (Mean topological dimension, Israel J. Math. 115 (2000), 1–24) for .     

Tame Automorphisms of Elementary Free Groups

We prove that the automorphism group of a finitely generated fully residually free group is tame.     

Twisted conjugacy classes in nilpotent groups

Let N be a finitely generated nilpotent group. We show that there is an algorithm that for any automorphism φ∈Aut(N) computes its Reidemeister number R(φ). It is proved that any free nilpotent group N_rc of rank r and class c belongs to class R_∞ if any of the following conditions holds: r=2 and c≥4; r=3 and c≥12; r≥4 and c≥2r.