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Abstract. Continuing Cavicchioli, Repovš, and Spaggiari's investigations into the cyclic presentations hx 1 ; : : : ; x n j x i x i Ck x i Cl D 1 .1 Ä i Ä n/i we determine when they are aspherical and when they define finite groups; in these cases we describe the groups' structures. In many cases we show that if the group is infinite then it contains a non-abelian free subgroup. Mathematics Subject Classification (2010). 20F05, 20F06.

THEOREM. Let G be a group and a,deG such that {\a\, \d\} =£{2, 3}. The equation at n dt~m = 1 (n, m e Z, n # m) has a solution over G. A.M.S. (1980) subject classification: 20E06.

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We study the 1-relator relative presentation 〈H, x|xaxbx−1c〉 where H is a group, a, b, c ∈ H, x ∉ H and b, c ≠ 1. We give necessary and sufficient conditions for this presentation to be aspherical apart from two outstanding special cases which remain open.