Amalgam decompositions for one-relator groups

Abstract

It is known that a one-relator group G with at least three generators admits a proper free product with amalgamation decomposition ( A★ B; C) with finitely generated factors. We call such a decomposition a Baumslag–Shalen decomposition. Little is known of the exact nature of the factors. Here we first prove that if the amalgamated subgroup is finitely presented, in particular free, then each factor is finitely presented. Using this we show that if G is a torsion-free one-relator group with Baumslag–Shalen decomposition ( A★ B; C) with C free, then each factor is homologically equivalent to either a free group or a one-relator group. Furthermore, sufficient conditions are obtained for G to be cyclically pinched if the factors are free groups.