Fractional Fibonacci groups with an odd number of generators

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 the 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 Q₈ and a finite cyclic group.