INTRODUCTION

The subject of this article is small cancellation techniques which when applied to certain Artin groups solves the word problem, the conjugacy problem and proves a conjecture of J. Tits for these groups. Our method is based on a new nonhomogeneous geometrical condition which is more flexible than the usual ones. This condition is a common generalization of the conditions C(4) and T(4) and the condition C(6). Recall that if a group G has a presentation which satisfies the condition C(6) then the corresponding van Kampen diagrams have the property that every inner region of them has at least 6 neighbours. Similarly, the condition C(4) and T(4) implies that every inner region has at least 4 neighbours and no inner vertex has valency 3. The condition C(6) corresponds in an obvious way to the regular essellation of the plane by hexagons and the conditions C(4) and T(4) corresponds to the regular tessellation of the plane by squares (see [6]). Our condition which we call condition W(6) applies to maps which in some places look like the hexagonal tessellation (see Fig. 1) and in some other places like the tessellation of the plane by squares (see Fig. 2) and in some more places look like the following two tessellations by pentagons (see Figs. 3a, 3b).