The structure of one-relator relative presentations and their centres

Abstract

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&#39; 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 Ĝ=<G,x_1,x_2,...,x_n | w=1> to the case n=1. It turns out that an &#34;n-variable&#34; group Ĝ can be constructed from similar &#34;one-variable&#34; groups using an explicit construction similar to wreath product. As an illustration, we prove that, for n>1, the centre of Ĝ is always trivial. For n=1, the centre of Ĝ is also almost always trivial; there are several exceptions, and all of them are known.