On amalgamated decompositions of finitely presented groups along splitting infinite cyclic groups (Q2639961)

Let G be a finitely presented group such that the Betti number of the first homology group \(H_ 1(G)\) is 1. Let x be an element of G mapped to a non-trivial element in an infinite cyclic group under some homomorphism. It is proved that there exists a unique decomposition \((G,x)=(G_ 1,x_ 1)*...*(G_ n,x_ n)\) with a finite number of factors and the amalgamation \(x=x_ 1=...=x_ n\) where \(x_ k\) is an element of infinite order of the group \(G_ k\), and every pair \((G_ k,x_ k)\) has no non-trivial such decomposition.