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2.4 LEMMA. Let M o be a simply connected submap of M with a connected interior which has no boundary edges on dM. If M satisfies W(6) and every simply connected proper submap satisfies 3tf v then dM 0 contains at least four vertices with valency 2 in M o .Proof. If <2) of Jfj contains at least four regions then it follows easily from 2f? x that the lemma holds. So let us check the remaining cases.Case 1, in which \®\ = 2. Let 2 = {D lt D^.Then i Mo (D x ) = i Mo (D 2 ) = 1, consequently each region D t , t = 1,2, contains at least two vertices with valency 2 in M o , on its boundary.Case 2 in which \3)\ = 3. Let 2) = {D lt D 2 ,D z }. Clearly, every region of 9

Let F denote a free group. An n-Engel word of F is an iterated commutator of the form [A, nB], where (A, B) is a pair of elements of F , which we here assume to generate a non-cyclic subgroup of F . This is the first of a series of three papers that yield a description of some of the properties -asphericity, word problem, local nilpotence -of a group defined by a suitable set of relators of the form [A i , nB i ], i = 1, 2, . . . , m (where n is fixed). The present paper develops and extends ideas introduced by the second author in [3], involving the use of cancellation diagrams, to provide the technical machinery for application in the subsequent companion papers.

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