# Fibonacci group

The Fibonacci group $F(2,m)$ has the presentation (cf. also Finitely-presented group; Presentation): where indices are taken modulo .

Fibonacci groups were introduced by J.H. Conway [a2] and are related to the Fibonacci numbers with inductive definition (with as initial ones).

Several combinatorial studies (see [a1] for references) answered some questions on , including their non-triviality and finiteness: is finite only for . H. Helling, A.C. Kim and J. Mennicke [a3] provided a geometrization of , by showing that the groups , , are the fundamental groups of certain closed orientable three-manifolds (so-called Fibonacci manifolds, denoted by ). See also Fibonacci manifold. In fact, for , , where is a closed hyperbolic three-manifold; , where is the Euclidean Hantzche–Wendt manifold; , with a lens space.

This and properties of the fundamental groups of these three-manifolds imply that are Noetherian groups, i.e. every finitely-generated subgroup of is finitely presented (cf. also Noetherian group). Since is an affine Riemannian manifold, is a torsion-free finite extension of . Due to hyperbolicity for (cf. also Hyperbolic group), the are torsion-free, their Abelian subgroups are cyclic (cf. also Cyclic group), there are explicit imbeddings , and the word and conjugacy problems are solvable for them (cf. also Group calculus; Identity problem). Also, the groups , , are arithmetic if and only if ; see [a3], [a4] and Arithmetic group.

There are several generalizations of Fibonacci groups, related to generalizations of Fibonacci numbers. D.L. Johnson [a5] has introduced the generalized Fibonacci groups (see [a9] for a survey) where indices are taken modulo . Another generalization of Fibonacci groups is due to C. Maclachlan [a7] (see [a8] for their geometrization):  Fractional Fibonacci groups were introduced by A.C. Kim and A. Vesnin in [a6] (which contains their geometrization as well):  