The cyclically presented groups with relators xi xi+k xi+l

Continuing Cavicchioli, Repovš, and Spaggiari's investigations into the cyclic presentations 〈x1,...,xn | x ixi+kxi+l D 1 (1 ≤ i ≤ n)〉 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. © European Mathematical Society.