Windmills and extreme 2-cells

TL;DR

This paper proves new conditions for the existence of extreme 2-cells in disc diagrams of 2-complexes, leading to results on group coherence, especially for certain classes of groups.

Contribution

It introduces conditions ensuring all minimal disc diagrams have extreme 2-cells, advancing the understanding of coherence in various group classes.

Findings

Extreme 2-cells exist under specific conditions in disc diagrams.

New coherence results for one-relator, small cancellation, and staggered presentation groups.

Perimeter-reduction techniques are effective for establishing coherence.

Abstract

In this article we prove new results about the existence of 2-cells in disc diagrams which are extreme in the sense that they are attached to the rest of the diagram along a small connected portion of their boundary cycle. In particular, we establish conditions on a 2-complex X which imply that all minimal area disc diagrams over X with reduced boundary cycles have extreme 2-cells in this sense. The existence of extreme 2-cells in disc diagrams over these complexes leads to new results on coherence using the perimeter-reduction techniques we developed in an earlier article. Recall that a group is called coherent if all of its finitely generated subgroups are finitely presented. We illustrate this approach by showing that several classes of one-relator groups, small cancellation groups and groups with staggered presentations are collections of coherent groups.