 
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 2cell 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 2cells. The first manner
in which our results augment the traditional results of small cancellation
