Indicable Groups and Endomorphic Presentations

TL;DR

This paper investigates the structure of finitely presented indicable groups, proving that certain subgroups have ascending finite endomorphic presentations and that such groups decompose as semidirect products with the integers.

Contribution

It establishes that finitely generated normal subgroups of finitely presented indicable groups have ascending finite endomorphic presentations, revealing their semidirect product structure.

Findings

Normal subgroups have ascending finite endomorphic presentations.

Indicable groups without free semigroups are semidirect products with Z.

Subgroup structure is characterized in terms of 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.