Abstract. We define small cancellation conditions W* and W representing non-positive curvature in cancellation diagrams. Each of these two conditions, which are dual to each other, generalizes the classical small cancellation conditions C(6), C(4) T(4), C(3) T(6). W also contains the small cancellation condition W(6) of Juhasz. Our main result is the solution of the conjugacy problem for groups with a presentation satisfying W* or W. Its proof uses the geometry of non-positively curved piecewise Euclidean complexes developed by Bridson. Following a sketch of Gersten, we also give a detailed proof of the solvability of the word problem for such groups by a quadratic isoperimetric inequality.

THE CONDITIONS W* AND W

Let P = 〈x1,…, xn |R1, …, Rm〉 be a finite presentation of the group G. We always assume that each relator Ri is cyclically reduced and no relator is the trivial word or a cyclic permutation of another relator or of the inverse of another relator. (By “relator” we always mean a denning relator of the presentation.) Let F be the free group on the generators. Kp denotes the standard 2–complex modeled on P.

The Whitehead graph Wp of Kp is the boundary of a regular neighborhood of the only vertex of Kp. For each generator xi of P it has two vertices +xi and –xi which correspond to the beginning and the end of the oriented loop labeled Xi in Kp.