Indicable Groups and Endomorphic Presentations

Abstract

In this note we look at presentations of subgroups of finitely presented groups with infinite cyclic quotients. We prove that if $H$ is a finitely generated normal subgroup of a finitely presented group $G$ with $G/H$ cyclic, then $H$ has ascending finite endomorphic presentation. It follows that any finitely presented indicable group without free semigroups has the structure of a semidirect product $H \rtimes \field{Z}$ where $H$ has finite ascending endomorphic presentation.