Klyachko's methods and the solution of equations over torsion-free groups (Q1920464)

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 products and HNN-extensions.