Introduction

It was believed for some time that most of the one-relator groups would be small cancellation groups. In this talk I would like to confirm this by showing that, with the exception of a small class of one-relator groups, they are small cancellation groups, and the groups in the exceptional class are small cancellation in a generalized sense, with one exception, where they are only small cancellation groups relative to a subgroup. These results are sufficient in order to regard one-relator groups as if they were small cancellation groups. In particular, we get a solution for the conjugacy problem for all one-relator groups. More precisely, it is shown that Schupp's solution to groups satisfying one of the geometrical small cancellation conditions [5] applies to one-relator groups with one exceptional class, where it is possible to reduce it to a combinatorial problem which cannot be solved by small cancellation theory but can be solved by a very simple combinatorial consideration [4].

The non-small cancellation groups

Let P = (X | R) be a one-relator presentation, R cyclically reduced. If P satisfies the condition C(6) then it is, by definition, small cancellation. So we shall assume that

P does not satisfy the condition C(6).

This means that there is a connected, simply connected, reduced in the sense of Lyndon, Schupp [5] diagram M over R (the symmetrical closure of R) which contains an inner region with less than six neighbours (i.e. a polygon with less than six edges).