Klyachko's methods and the solution of equations over torsion-free groups
type: Item
instance of: scholarly article
title: Klyachko's methods and the solution of equations over torsion-free groups
zbMATH Open document ID: 0861.20029
author: Roger Fenn
author: Colin Rourke
published in: L'Enseignement Mathématique. 2e Série
publication date: 1997-05-13T00:00:00Z
review text: If \(G\) is a torsion-free group, \(G*\langle t\rangle\) the free product of \(G\) with the infinite cyclic group \(\langle t\rangle\) and \(w\) an element of \(G*\langle t\rangle\) not belonging to \(G\), then there is the question whether the natural map \(G\to G*\langle t\rangle/\langle\langle w\rangle\rangle\) is injective. Here \(\langle\langle w\rangle\rangle\) denotes the normal closure of \(w\) in \(G*\langle t\rangle\). It was proved by \textit{A. A. Klyachko} in 1993 [Commun. Algebra 21, No. 7, 2555-2575 (1993; Zbl 0788.20017)] that the above map is injective if the exponent sum of \(t\) in \(w\) is \(\pm1\) and if it is not \(\pm1\) then the quotient \(G*\langle t\rangle/\langle\langle w\rangle\rangle\) has a non-trivial cyclic quotient. This proves in the affirmative the Kervaire conjecture for torsion free groups which says that if \(G\) is a non-trivial torsion-free group, then \(G*\langle t\rangle/\langle\langle w\rangle\rangle\) is non-trivial. This problem is relative to the equation \(w=1\). Here if \(w=w(t)\), then \(w=1\) is an equation in the variable \(t\). We say that \(w=1\) has a solution over \(G\) if there exists a group \(\overline G\) containing \(G\) as a subgroup and element \(x\in\overline G\), such that \(w(x)=1\) and it is easy to see that \(w=1\) has a solution over \(G\) if the natural map \(G\to G*\langle t\rangle/\langle\langle w\rangle\rangle\) is injective. In this paper the authors give a clear exposition of the above mentioned paper of Klyachko and state and explain in full detail the results of that paper. Then they use the methods of Klyachko of traffic flows and complete crushes along the boundary of cells of a 2-sphere to prove that if \(w\) is an amenable word of \(G*\langle t\rangle-G\), then the equation \(w=1\) has a solution over \(G\), where \(G\) is nontrival torsion-free and get as a corollary the positive answer of the Kervaire conjecture for non-trivial torsion-free groups. They give also applications to free p…
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 915929
zbMATH Keywords: Kervaire conjecture
zbMATH Keywords: torsion free groups
zbMATH Keywords: equations
zbMATH Keywords: solutions
zbMATH Keywords: free products
zbMATH Keywords: HNN-extensions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 915929