Isomorphism theorems for classes of cyclically presented groups

We consider two multi-parameter classes of cyclically presented groups, introduced by Cavicchioli, Repov s, 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

An Investigation Into the Cyclically Presented Groups with Length Three Positive Relators

We continue research into the cyclically presented groups with length three positive relators. We study small cancelation conditions, SQ-universality, and hyperbolicity, 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

Counting isomorphism classes of groups of Fibonacci type with a prime power number of generators

Cavicchioli, O'Brien, and Spaggiari studied the number of isomorphism classes of irreducible groups of Fibonacci type as a function σ(n) of the number of generators n. In the case n=pl, where p is prime and l≥1, n≠2,4, they conjectured a function C(pl), that is polynomial in p, for the value of σ(pl). We prove that C(pl) is an upper bound for σ(pl). We introduce a function τ(n) for the number of abelianised groups and conjecture a function D(pl), that is polynomial in p, for the value of τ(pl), when pl≠2,4,5,7,8,13,23. We prove that D(pl) is an upper bound for τ(pl). We pose three questions that ask if particular pairs of groups with common abelianisations are non-isomorphic. We prove that if τ(pl)=D(pl) and each of these questions has a positive answer then σ(pl)=C(pl)