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In this article we prove new results about the existence of 2-cells in discdiagrams which are extreme in the sense that they are attached to the rest ofthe diagram along a small connected portion of their boundary cycle. Inparticular, we establish conditions on a 2-complex X which imply that allminimal area disc diagrams over X with reduced boundary cycles have extreme2-cells in this sense. The existence of extreme 2-cells in disc diagrams overthese complexes leads to new results on coherence using the perimeter-reductiontechniques we developed in an earlier article. Recall that a group is calledcoherent if all of its finitely generated subgroups are finitely presented. Weillustrate this approach by showing that several classes of one-relator groups,small cancellation groups and groups with staggered presentations arecollections of coherent groups.

Windmills and extreme 2-cells