 
Summary: WINDMILLS AND EXTREME 2CELLS
JON MCCAMMOND 1 AND DANIEL WISE 2
Abstract. In this article we prove new results about the existence of 2cells 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 par
ticular, we establish conditions on a 2complex X which imply that all minimal
area disc diagrams over X with reduced boundary cycles have extreme 2cells
in this sense. The existence of extreme 2cells in disc diagrams over these
complexes leads to new results on coherence using the perimeterreduction
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 onerelator groups,
small cancellation groups and groups with staggered presentations are collec
tions of coherent groups.
In this article we prove some new results about the existence of extreme 2cells in
disc diagrams which lead to new results on coherence. In particular, we combine the
diagram results shown here with the theorems from [3] to establish the coherence
of various classes of onerelator groups, small cancellation groups, and groups with
relatively staggered presentations. The article is organized as follows: § 1 contains
background definitions, § 2 recalls how extreme 2cells lead to perimeter reductions
