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WINDMILLS AND EXTREME 2-CELLS JON MCCAMMOND 1 AND DANIEL WISE 2
 

Summary: WINDMILLS AND EXTREME 2-CELLS
JON MCCAMMOND 1 AND DANIEL 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 par-
ticular, 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 collec-
tions 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 [3] 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: 1 contains
background definitions, 2 recalls how extreme 2-cells lead to perimeter reductions

  

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

 

Collections: Mathematics