where indices are taken modulo .
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.
Fractional Fibonacci groups were introduced by A.C. Kim and A. Vesnin in [a6] (which contains their geometrization as well):
|[a1]||C.M. Campbell, "Topics in the theory of groups" , Notes on Pure Math. , I , Pusan Nat. Univ. (1985)|
|[a2]||J.H. Conway, "Advanced problem 5327" Amer. Math. Monthly , 72 (1965) pp. 915|
|[a3]||H. Helling, A.C. Kim, J. Mennicke, "A geometric study of Fibonacci groups" J. Lie Theory , 8 (1998) pp. 1–23|
|[a4]||H.M. Hilden, M.T. Lozano, J.M. Montesinos, "The arithmeticity of the figure-eight knot orbifolds" B. Apanasov (ed.) W. Neumann (ed.) A. Reid (ed.) L. Siebenmann (ed.) , Topology'90 , de Gruyter (1992) pp. 169–183|
|[a5]||D.L. Johnson, "Extensions of Fibonacci groups" Bull. London Math. Soc. , 7 (1974) pp. 101–104|
|[a6]||A.C. Kim, A. Vesnin, "The fractional Fibonacci groups and manifolds" Sib. Math. J. , 38 (1997) pp. 655–664|
|[a7]||C. Maclachlan, "Generalizations of Fibonacci numbers, groups and manifolds" , Combinatorial and Geometric Group Theory (1993) , Lecture Notes , 204 , London Math. Soc. (1995) pp. 233–238|
|[a8]||C. Maclachlan, A.W. Reid, "Generalized Fibonacci manifolds" Transformation Groups , 2 (1997) pp. 165–182|
|[a9]||R.M. Thomas, "The Fibonacci groups revisited" C.M. Campbell (ed.) E.F. Robertson (ed.) , Groups II (St. Andrews, 1989) , Lecture Notes , 160 , London Math. Soc. (1991) pp. 445–456|
Fibonacci group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Fibonacci_group&oldid=21713