Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ,1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homotopy theory.     

Automorphism groups of nilpotent groups

We construct a full class of nilpotent groups of class 2 of an arbitrary infinite cardinality . Their centers, commutator subgroups and factors modulo the center will be the same and a homogeneous direct sum of a group of rank 1 or 2. Their automorphism groups will coincide and the factor group modulo the stabilizer could be an arbitrary group of size $\leqq$ .     

Holonomy groups of Bieberbach groups with finite outer automorphism groups

This work was supported by the DFG and by a Polish grant BW/5100-5-0116-3.     

Isometry groups of three-dimensional Lie groups

Annals of Global Analysis and Geometry - We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we...     

Endomorphisms of automorphism groups of free groups

It is proved that any non-trivial endomorphism of an automorphism group AutF_n of a free group F_n, for n 3, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of AutF₂ is an automorphism, or else a homomorphism onto one of the groups S₃, D₈, Z₂ × Z₂, Z₂, or (Z₂ × Z₂). A non-trivial homomorphism of AutF_n into AutF_m, for n 3, m 2, and n > m, is a homomorphism onto Z₂ with kernel SAutF_n. As a consequence, we obtain that AutF_n is co-Hopfian.Supported by RFBR grant No. 02-01-00293 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 211–237, March–April, 2005.     

Quotient groups of locally graded groups and groups of certain Kurosh-Chernikov classes

We establish the validity of the inclusion G/N∈ X for groups G ∈ X under certain restrictions on N ⊴ G, where X is one of the following classes, the class of locally graded groups, the class of RI-groups, or the class for a fixed group variety . Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1545–1553, November, 1998.     

Automorphism groups of free metabelian nilpotent groups

Published in Algebra i Logika, Vol. 29, No. 6, pp. 746–751, November–December, 1990.     

Holonomy groups of five dimensional Bieberbach groups

Let Γ be a Bieberbach group—that is a torsion free crystallographic group. In this paper we give a list of the isomorphism types of all holonomy groups of five-dimensional Bieberbach groups. An application to the problem of estimating the covolume of a discrete group of orientation-preserving isometries of hyperbolic 6-space is also given. This work was supported by Polish grant nr.0627/P3/93/04