Windmills and extreme $2$-cells

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.