Summary: WINDMILLS AND EXTREME 2­CELLS
JONATHAN P. MCCAMMOND 1 AND DANIEL T. 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 particular, 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 collections 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. Section 1
contains background definitions, Sections 2 introduces the concept of a windmill,

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics