Windmills and extreme $2$-cells

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 [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 and to coherent fundamental groups, § 3 introduces the concept of a windmill, § 4 uses windmills to prove that extreme 2-cells exist, and finally § 5 uses extreme 2-cells to prove that various groups are coherent. For instance, we obtain the following special case of Corollary 5.12: Corollary 0.1. Let G = 〈a1, . . . , ar, t |W 〉 where W has the form t W1t W2 . . . t Wk, N is arbitrary, and for each i, i is a nonzero integer and Wi is a reduced word in the ai. Suppose that {W1,W2, . . . ,Wk} freely generate a subgroup of the free group 〈a1, . . . , ar | −〉. Then G is coherent. 1. Basic Definitions In this section we review some basic definitions about 2-complexes and diagrams. Definition 1.1 (Combinatorial maps and complexes). A map Y → X between CW complexes is combinatorial if its restriction to each open cell of Y is a homeomorphism onto an open cell of X. A CW complex X is combinatorial provided Date: September 7, 2009. 2000 Mathematics Subject Classification. 20F06,20F67,57M07.