Fractional Fibonacci groups with an odd number of generators

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 Fk/l(n). We show that for odd n the group Fk/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 Fk/l(n) (where n is odd) paying particular attention to the groups Fk(n) and Fk/l(3), and observe consequences concerning the asphericity of relative presentations of their shift extensions. We show that if Fk(n) (where n is odd) and Fk/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.