Fibonacci group

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

\begin{equation*} F ( 2 , m ) = \langle x _ { 1 } , \dots , x _ { m } | x _ { i } x _ { i + 1} = x _ { i + 2} \rangle, \end{equation*}

where indices are taken modulo $m$.

Fibonacci groups were introduced by J.H. Conway [a2] and are related to the Fibonacci numbers with inductive definition $a _ { i } + a _ { i + 1 } = a _ { i + 2 }$ (with $a _ { 1 } = a _ { 2 } = 1$ as initial ones).

Several combinatorial studies (see [a1] for references) answered some questions on $F ( 2 , m )$, including their non-triviality and finiteness: $F ( 2 , m )$ is finite only for $m = 1,2,3,4,5,7$. H. Helling, A.C. Kim and J. Mennicke [a3] provided a geometrization of $F ( 2 , m )$, by showing that the groups $F ( 2,2 n )$, $n \geq 2$, are the fundamental groups of certain closed orientable three-manifolds (so-called Fibonacci manifolds, denoted by $M _ { n }$). See also Fibonacci manifold. In fact, for $n \geq 4$, $F ( 2,2 n ) = \pi _ { 1 } ( M _ { n } )$, where $M _ { n }$ is a closed hyperbolic three-manifold; $F ( 2,6 ) = \pi _ { 1 } ( M _ { 3 } )$, where $M _ { 3 }$ is the Euclidean Hantzche–Wendt manifold; $F ( 2,4 ) = \pi _ { 1 } ( L ( 5,2 ) )$, with $L ( 5,2 )$ a lens space.

This and properties of the fundamental groups of these three-manifolds imply that $F ( 2,2 n )$ are Noetherian groups, i.e. every finitely-generated subgroup of $F ( 2,2 n )$ is finitely presented (cf. also Noetherian group). Since $M _ { 3 }$ is an affine Riemannian manifold, $F ( 2,6 )$ is a torsion-free finite extension of $\mathbf{Z} ^ { 3 }$. Due to hyperbolicity for $n \geq 4$ (cf. also Hyperbolic group), the $F ( 2,2 n )$ are torsion-free, their Abelian subgroups are cyclic (cf. also Cyclic group), there are explicit imbeddings $F ( 2,2 n ) \subset \operatorname { PSL } _ { 2 } ( {\bf C} )$, and the word and conjugacy problems are solvable for them (cf. also Group calculus; Identity problem). Also, the groups $F ( 2,2 n )$, $n \geq 4$, are arithmetic if and only if $n = 4,5,6,8,12$; 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)

\begin{equation*} F ( r , m ) = ( x _ { 1 } , \dots , x _ { m } | x _ { i } \dots x _ { i + r - 1} = x _ { i + r } ), \end{equation*}

where indices are taken modulo $m$. Another generalization of Fibonacci groups is due to C. Maclachlan [a7] (see [a8] for their geometrization):

\begin{equation*} F ^ { k } ( 2 , m ) = \end{equation*}

\begin{equation*} = \langle x _ { 1 } , \dots , x _ { m } | x _ { i } x ^ { k _ { i + 1} } = x _ { i + 2 } ; \text { indices } ( \operatorname { mod } m ) \rangle. \end{equation*}

Fractional Fibonacci groups were introduced by A.C. Kim and A. Vesnin in [a6] (which contains their geometrization as well):

\begin{equation*} F ^ { k / l } ( 2 , m ) = \end{equation*}

\begin{equation*} = \left\langle x _ { 1 } , \ldots , x _ { m } | x ^ { l_i } x ^ { k _ { i + 1} } = x ^ { l _ { i + 2 } } ; \text { indices } ( \operatorname { mod } m ) \right\rangle. \end{equation*}

References

[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