Small cancellation theory with a weakened small cancellation hypothesis. I: The basic theory (Q1082443)

This is the first in a series of papers in which the author develops a small-cancellation theory under a hypothesis W(4) which is weaker than the condition C(4) \& T(4). One of the motivations for this is to solve the conjugacy problem for certain classes of one-relator groups, which satisfy W(4) but not any of the standard small cancellation conditions. Roughly speaking, the C and T conditions relate to the degrees of regions and vertices respectively in van Kampen diagrams. The W condition is a mixture of the two. It consists of a sequence of conditions on the degrees of vertices around a given region, depending on the degree of the region. In the present paper the basic theory is developed, and a version of Greendlinger's Lemma is proved.