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The structure of one-relator relative presentations and their centres. (English) Zbl 1203.20030
Let $$G$$ be a group. A group given by a one-relator relative presentation over $$G$$ is a group $$\widehat G=\langle G,x_1,\dots,x_n\mid w=1\rangle=(G*F(x_1,\dots,x_n))/\langle\langle w\rangle\rangle$$ where $$x_1,\dots,x_n$$ are letters (not belonging to $$G$$) and $$w$$ is a word in the alphabet $$G\cup\{x_1^{\pm 1},\dots,x_n^{\pm 1}\}$$.
In a series of papers the author investigates such groups $$\widehat G$$ under the hypothesis that $$G$$ is torsion-free and that the word formed from $$w$$ by erasing coefficients belonging to $$G$$ is not a proper power. He continues such investigations here. “The main result of this paper (Theorem 3) shows how to reduce the study of the groups $$\widehat G$$ to the case $$n=1$$. It turns out that the ‘$$n$$-variable’ group $$\widehat G$$ can be obtained from similar ‘one-variable’ groups by using an explicit construction that involves free iterated amalgamated products (see Section 3) and amalgamated semidirect products (see Section 4). This structural theorem (Theorem 3) may have a number of applications. One application is considered in this paper. Namely, we study the centre of the group $$\widehat G$$.”
In the case $$n=1$$ where $$w$$ is unimodular (i.e. the exponent sum of $$x_1$$ in $$w$$ is 1) the author has previously shown that $$\widehat G$$ inherits some properties of $$G$$. For instance, if $$G$$ is non-trivial, or non-Abelian, or torsion-free, or non-simple, or is a group that satisfies the Tits alternative, then so is $$\widehat G$$ (respectively). As an application of the main result another property of this kind is established in Theorem 1: the centre of $$\widehat G$$ is trivial except when either (1) $$w\equiv gtg'$$ where $$g,g'\in G$$ and the centre of $$G$$ is non-trivial; or (2) $$G$$ is cyclic and $$\widehat G$$ is a one-relator group with non-trivial centre.
Passing to the multivariable case a new generalisation of the notion of unimodularity is introduced and is used for Theorem 2 which generalizes Theorem 1. Its corollary (Corollary 1) is a multivariable analogue of Theorem 1 showing, in particular, that for $$n\geq 2$$ the centre of $$\widehat G$$ is trivial.
“These results on the centre of $$\widehat G$$ are not surprising. However, they easily imply the Kervaire-Laudenbach conjecture for torsion-free groups [A. A. Klyachko, Commun. Algebra 21, No. 7, 2555-2575 (1993; Zbl 0788.20017)], i.e., the non-triviality of each group of the form $$\langle H,t\mid w=1\rangle$$ where $$H$$ is a non-trivial torsion-free group and $$w$$ is any word in the alphabet $$H\cup\{t^{\pm 1}\}$$.”

##### MSC:
 20F05 Generators, relations, and presentations of groups 20F06 Cancellation theory of groups; application of van Kampen diagrams 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20E22 Extensions, wreath products, and other compositions of groups 20E07 Subgroup theorems; subgroup growth 20F70 Algebraic geometry over groups; equations over groups 57M07 Topological methods in group theory
Zbl 0788.20017
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