Fractional Fibonacci groups with an odd number of generators

TL;DR

This paper explores the properties of Fractional Fibonacci groups with an odd number of generators, revealing their geometric, torsion, and asphericity characteristics, and classifying certain cases as direct products involving quaternion groups.

Contribution

It extends known results for Fibonacci groups to fractional variants, analyzing their geometric structures, torsion properties, and classifying specific cases as quaternionic products.

Findings

F^{k/l}(n) with odd n are not fundamental groups of orientable hyperbolic 3-orbifolds.

Identifies conditions under which these groups contain torsion.

Shows certain non-cyclic groups are isomorphic to quaternion times cyclic groups.

Abstract

The Fibonacci groups $F(n)$ are known to exhibit significantly different behaviour depending on the parity of $n$. We extend known results for $F(n)$ for odd $n$ to the family of Fractional Fibonacci groups $F^{k/l}(n)$. We show that for odd $n$ the group $F^{k/l}(n)$ is not the fundamental group of an orientable hyperbolic 3-orbifold of finite volume. We obtain results concerning the existence of torsion in the groups $F^{k/l}(n)$ (where $n$ is odd) paying particular attention to the groups $F^k(n)$ and $F^{k/l}(3)$, and observe consequences concerning asphericity of relative presentations of their shift extensions. We show that if $F^{k}(n)$ (where $n$ is odd) and $F^{k/l}(3)$ are non-cyclic 3-manifold groups then they are isomorphic to the direct product of the quaternion group $Q_8$ and a finite cyclic group.