FANS AND LADDERS IN SMALL CANCELLATION THEORY Summary: FANS AND LADDERS IN SMALL CANCELLATION THEORY JONATHAN P. MCCAMMOND1 AND DANIEL T. WISE2 1. Introduction Small cancellation theory and its various generalizations have proven to be powerful tools in the study of infinite groups, particularly for the con- struction 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 Collections: Mathematics