Fans and Ladders in Small Cancellation Theory

Small cancellation theory and its various generalizations have proven to be powerful tools in the study of infinite groups, particularly for the construction of examples of groups exhibiting specific properties. In this article we derive statements which are similar to but significantly stronger than the usual small cancellation formulations. These stronger results are presented using the notion of a fan, which is introduced here for the first time. The main geometric conclusion in small cancellation theory is essentially that disc diagrams contain a 2-cell most of which lies on the very outside of the diagram, as illustrated on the left in Figure 1. In studying this situation, we found that it can be strengthened. Specifically, we show that disc diagrams satisfying small cancellation conditions have a sequence of consecutive cells all of which lie near the outside of the diagram. We call the union of these cells in the diagram a fan. The diagram on the right in Figure 1 contains a fan which is the union of four 2-cells. The first manner in which our results augment the traditional results of small cancellation theory is that when the small-cancellation conditions are sufficiently strict, the disc diagram will contain fans consisting of longer and longer chains of 2-cells. Our second contribution to this subject is a classification result for disc diagrams. We show that either a disc diagram is small and round and contains fans in all directions, or it is long and relatively thin like a ladder and contains disjoint fans at each end, or in the generic case, it contains at least three sharp turns along three disjoint fans. This trichotomy is analogous to the trichotomy for finite trees: A finite tree either consists of a single point, or it is homeomorphic to a real interval, or it has at least three leaves.