Dion Gildenhuys

CSA-groups and separated free constructions

A group

A Kurosh subgroup theorem for free pro--products of pro--groups

G

THE WORD PROBLEM FOR SOME VARIETIES OF SOLVABLE LIE ALGEBRAS

is said to be a {\it CSA}-group if all maximal abelian subgroups of

Profinite groups and boolean graphs

G

Classification of soluble groups of cohomological dimension two

are malnormal. The class of CSA groups is of interest because it contains torsion-free hyperbolic groups, groups acting freely on

Algorithmically insoluble problems about finitely presented solvable groups, lie and associative algebras. I

\Lambda

Free pro-C-groups

-trees and groups with the same existential theory as free groups. CSA groups are also very closely related to the study of residually free groups and tensor completions. In this paper we investigate which free constructions (amalgamated products and HNN extensions) over CSA groups are again CSA. The results are applied, in particular, to show that a torsion-free one-relator group is CSA if and only if it does not contain nonabelian metabelin Baumslag-Solitar groups and the direct product of the free group of rank 2 and the infinite cyclic group.

Equational completion, model induced triples and pro-objects

Let C be a class of finite groups, closed under finite products, subgroups and homomorphic images. In this paper we define and study free pro-e- products of pro-e- groups indexed by a pointed topological space. Our main result is a structure theorem for open subgroups of such free products along the lines of the Kurosh subgroup theorem for abstract groups. As a consequence we obtain

On the cohomology of soluble groups II

The word problem is said to be solvable in a variety of Lie algebras if it is solvable in every algebra, finitely presented in this variety. Let denote the variety of (2-step nilpotent)-by-abelian Lie algebra and the variety of abelian-by-(2-step nilpotent) Lie algebras. It is proved that the word problem is unsolvable in the “interval” of varieties containing the variety (of centre-by- Lie algebras over a field of characteristic zero), and contained in the variety .