The cyclically presented groups with relators $x_i{x}_{i+k}{x}_{i+l}$

Abstract

Continuing Cavicchioli, Repovš, and Spaggiari’s investigations into the cyclic presentations $⟨x_1,\dots ,x_n|x_i{x}_{i+k}{x}_{i+l}=1(1\le i\le n)⟩$ 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.