Coherence, local quasiconvexity, and the perimeter of 2-complexes

TL;DR

This paper introduces a new criterion based on the perimeter of maps between 2-complexes to determine group coherence and local quasiconvexity, with applications to various classes of groups including one-relator and small cancellation groups.

Contribution

It provides a novel perimeter-based criterion for group coherence and local quasiconvexity, enabling efficient subgroup presentation computation and broad applicability.

Findings

Groups with the new criterion are coherent and locally quasiconvex

Finitely generated subgroups can be presented in quadratic time

Many small cancellation groups are shown to be coherent and locally quasiconvex

Abstract

A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a map between two finite 2-complexes which is introduced here. In the groups to which this theory applies, a presentation for a finitely generated subgroup can be computed in quadratic time relative to the sum of the lengths of the generators. For many of these groups we can show in addition that they are locally quasiconvex. As an application of these results we prove that one-relator groups with sufficient torsion are coherent and locally quasiconvex and we give an alternative proof of the coherence and local quasiconvexity of certain 3-manifold groups. The main application is to establish the coherence and local quasiconvexity of many small cancellation groups.