Groups of Fibonacci type revisited

This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway?s Fibonacci groups, the Sieradski groups, and the Gilbert-Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repov?s, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labelled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups

Fibonacci type presentations and 3-manifolds

We study the cyclic presentations with relators of the form xixi+mx?1 i+k and the groups they define. These ?groups of Fibonacci type? were intro- duced by Johnson and Mawdesley and they generalize the Fibonacci groups F(2, n) and the Sieradski groups S(2, n). With the exception of two groups, we classify when these groups are fundamental groups of 3-manifolds, and it turns out that only Fibonacci, Sieradski, and cyclic groups arise. Using this classification, we completely classify the presentations that are spines of 3-manifolds, answering a question of Cavicchioli, Hegenbarth, and Repov?s. When n is even the groups F(2, n), S(2, n) admit alternative cyclic presenta- tions on n/2 generators. We show that these alternative presentations also arise as spines of 3-manifolds

Efficient Finite Groups Arising in the Study of Relative Asphericity

We study a class of two-generator two-relator groups, denoted Jn(m, k), that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature as finite groups of intriguing orders. Here we find infinite families of non-elementary virtually free groups and of finite metabelian non-nilpotent groups, for which we determine the orders. All Mersenne primes arise as factors of the orders of the non-metacyclic groups in the class, as do all primes from other conjecturally infinite families of primes. We classify the finite groups up to isomorphism and show that our class overlaps and extends a class of groups Fa,b,c with trivalent Cayley graphs that was introduced by C.M.Campbell, H.S.M.Coxeter, and E.F.Robertson. The theory of cyclically presented groups informs our methods and we extend part of this theory (namely, on connections with polynomial resultants) to ?bicyclically presented groups? that arise naturally in our analysis. As a corollary to our main results we obtain new infinite families of finite metacyclic generalized Fibonacci groups

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

We study a class M of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analysing their shift extensions we show that the groups in the class M are are coherent, sub-group separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all meta-cyclic. Many of the groups in M are virtually free, some are free products of metacyclic groups and free groups, and some have geometric dimension two. We classify the finite groups that occur in M, giving extensive details about the metacyclic structures that occur, and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three. We show that any finite group in the class M that has fixed point free shift automorphism must be cyclic

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)

Cyclically presented groups as Labelled Oriented Graph groups

We use results concerning the Smith forms of circulant matrices to identify when cyclically presented groups have free abelianisation and so can be Labelled Oriented Graph (LOG) groups. We generalize a theorem of Odoni and Cremona to show that for a fixed defining word, whose corresponding representer polynomial has an irreducible factor that is not cyclotomic and not equal to ±t, there are at most finitely many n for which the corresponding n-generator cyclically presented group has free abelianisation. We classify when Campbell and Robertson's generalized Fibonacci groups H(r,n,s) are LOG groups and when the Sieradski groups are LOG groups. We prove that amongst Johnson and Mawdesley's groups of Fibonacci type, the only ones that can be LOG groups are Gilbert-Howie groups H(n,m). We conjecture that if a Gilbert-Howie group is a LOG group, then it is a Sieradski group, and prove this in certain cases (in particular, for fixed m, the conjecture can only be false for finitely many n). We obtain necessary conditions for a cyclically presented group to be a connected LOG group in terms of the representer polynomial and apply them to the Prishchepov groups

Hyperbolicity of T(6) Cyclically Presented Groups

We consider groups defined by cyclic presentations where the defining word has length three and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known results, this completely classifies the hyperbolic T(6) cyclically presented groups

Hyperbolic groups of Fibonacci type and T(5) cyclically presented groups

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H(9, 4),H(9, 7). We show that if H(9, 4) is torsion-free then it is not hyperbolic. We consider the class of T(5) cyclically presented groups and classify the (non-elementary) hyperbolic groups and show that the Tits alternative holds

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

Aspherical Relative Presentations All Over Again

The concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged. In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for proving non-asphericity. We give a detailed survey of results concerning one-relator relative presentations where the relator has free product length four