17 research outputs found
The structure of one-relator relative presentations and their centres
Suppose that G is a nontrivial torsion-free group and w is a word in the
alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\} such that the word w' obtained from
w by erasing all letters belonging to G is not a proper power in the free group
F(x_1,...,x_n). We show how to reduce the study of the relative presentation
\^G= to the case n=1. It turns out that an
"n-variable" group \^G can be constructed from similar "one-variable" groups
using an explicit construction similar to wreath product. As an illustration,
we prove that, for n>1, the centre of \^G is always trivial. For n=1, the
centre of \^G is also almost always trivial; there are several exceptions, and
all of them are known.Comment: 15 pages. A Russian version of this paper is at
http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm . V4:
the intoduction is rewritten; Section 1 is extended; a short introduction to
Secton 5 is added; some misprints are corrected and some cosmetic
improvements are mad
Unions of cockcroft two--complexes
A combinatorial group-theoretic hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This hypothesis has a component that can be expressed in terms of the second homology of groups. The hypothesis is applied to the study of the third homology of groups given by generators and relators
The Homology of Peiffer Products of Groups
The Peiffer product of groups first arose in work of J.H.C. Whitehead on the structure of relative homotopy groups, and is closely related to problems of asphericity for two-complexes. We develop algebraic methods for computing the second integral homology of a Peiffer product. We show that a Peiffer product of superperfect groups is superperfect, and determine when a Peiffer product of cyclic groups has trivial second homology. We also introduce a double wreath product as a Peiffer product