Summary: WINDMILLS AND EXTREME 2-CELLS
JONATHAN P. MCCAMMOND 1 AND DANIEL T. WISE 2
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.
In this article we prove some new results about the existence of extreme 2-cells in
disc diagrams which lead to new results on coherence. In particular, we combine the
diagram results shown here with the theorems from  to establish the coherence
of various classes of one-relator groups, small cancellation groups, and groups with
relatively staggered presentations. The article is organized as follows. Section 1
contains background definitions, Sections 2 introduces the concept of a windmill,