Cyclically presented groups as Labelled Oriented Graph groups

TL;DR

This paper investigates conditions under which cyclically presented groups can be classified as Labelled Oriented Graph groups, using algebraic and polynomial methods, and provides classifications and conjectures about specific group families.

Contribution

It generalizes existing theorems to classify when cyclically presented groups are LOG groups and analyzes specific families like Fibonacci and Sieradski groups.

Findings

Identifies conditions for free abelianisation of cyclically presented groups.

Classifies when certain Fibonacci-type groups are LOG groups.

Proposes conjectures relating Gilbert-Howie and Sieradski groups.

Abstract

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 $\pm t$, 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.