We continue research into the cyclically presented groups with length three positive relators. We study small cancellation conditions and SQ-universality, we obtain the Betti numbers of the groups' abelianisations, we calculate the orders of the abelianisations of some groups, and we study isomorphism classes of the groups. Through computational experiments we assess how effective the abelianisation is as an invariant for distinguishing non-isomorphic groups.

We consider two multi-parameter classes of cyclically presented groups, introduced by Cavicchioli, Repovš and Spaggiari, that contain many previously considered families of cyclically presented groups of interest both for their algebraic and for their topological properties. Building on results of Bardakov and Vesnin, O’Brien and the previously named authors, we prove theorems that establish isomorphisms of groups within these families.

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