Free Subgroups of Small Cancellation Groups

with the property that for any pair r, s of elements of R either r = s or there is very little free cancellation in forming the product rs. The classical example of such a group is the fundamental group of a closed orientable 2-manifold of genus k. The study of this group has given rise both to the theory of one-relator groups, initiated by Magnus ([3]), and to the theory of small cancellation groups, initiated by Tartakovskii ([8]). Now it turns out that one-relator groups and small cancellation groups have some similar properties. Thus for example they often have trivial centre and in many cases the only torsion elements are those 'immediately visible' from the presentation. In view of Magnus's Freiheitssatz it is natural to consider whether or not small cancellation groups contain free subgroups of rank two. Soldatova ([7]) has shown that this is often the case. Our aim is to generalize as far as possible some of her results to a wider class of small cancellation groups. In general our notation and terminology will be that introduced by Lyndon in [2] (see also Schupp ([5], [6])). We recall the following basic definitions. Let J? be a symmetrized set of words, i.e. let R satisfy the conditions (i) if r 6 R, then r is cyclically freely reduced, (ii) if r e R, then every cyclic permutation of both r and r~ also lies in R. A word a; is a piece in R if there exist words r,s e R such that r = xr', s = xs' with r' and s' distinct words. We say that 0 = (a,b,c,...; r = 1, (r e R)) satisfies condition C(k) if R is a symmetrized set of words such that no element of R is a product of fewer than k pieces. We also say that G satisfies condition T(q), q ^ 4, if, for any h, 3 ^ h < q, and words rx, r2,..., rh e R such that in no pair i>i+i (including rh,rx) are the members inverse to one another, then at least one of the words r1r2,r2r3, ..., PA-ITA. rhr1 is freely reduced.