The structure of one-relator relative presentations and their centres.

*(English)*Zbl 1203.20030Let \(G\) 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 \(H\cup\{t^{\pm 1}\}\).”

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 \(H\cup\{t^{\pm 1}\}\).”

Reviewer: Gerald Williams (Colchester)

##### 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 |

##### Citations:

Zbl 0788.20017
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\textit{A. A. Klyachko}, J. Group Theory 12, No. 6, 923--947 (2009; Zbl 1203.20030)

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