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

<jats:p> Continuing Cavicchioli, Repovš, and Spaggiari’s investigations into the cyclic presentations <jats:inline-formula> <jats:tex-math>\langle x_1, \ldots , x_n | x_ix_{i+k}x_{i+l} = 1 (1 ≤ i ≤ n) \rangle</jats:tex-math> </jats:inline-formula> 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. </jats:p>