View source for Fibonacci group

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300702.png" /></td> </tr></table>

where indices are taken modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300703.png" />.

Fibonacci groups were introduced by J.H. Conway [[#References|[a2]]] and are related to the [[Fibonacci numbers|Fibonacci numbers]] with inductive definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300704.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300705.png" /> as initial ones).

Several combinatorial studies (see [[#References|[a1]]] for references) answered some questions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300706.png" />, including their non-triviality and finiteness: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300707.png" /> is finite only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300708.png" />. H. Helling, A.C. Kim and J. Mennicke [[#References|[a3]]] provided a geometrization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300709.png" />, by showing that the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007011.png" />, are the fundamental groups of certain closed orientable three-manifolds (so-called Fibonacci manifolds, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007012.png" />). See also [[Fibonacci manifold|Fibonacci manifold]]. In fact, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007015.png" /> is a closed hyperbolic three-manifold; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007016.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007017.png" /> is the Euclidean Hantzche–Wendt manifold; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007018.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007019.png" /> a [[Lens space|lens space]].

This and properties of the fundamental groups of these three-manifolds imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007020.png" /> are Noetherian groups, i.e. every finitely-generated subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007021.png" /> is finitely presented (cf. also [[Noetherian group|Noetherian group]]). Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007022.png" /> is an affine [[Riemannian manifold|Riemannian manifold]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007023.png" /> is a torsion-free finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007024.png" />. Due to hyperbolicity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007025.png" /> (cf. also [[Hyperbolic group|Hyperbolic group]]), the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007026.png" /> are torsion-free, their Abelian subgroups are cyclic (cf. also [[Cyclic group|Cyclic group]]), there are explicit imbeddings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007027.png" />, and the word and conjugacy problems are solvable for them (cf. also [[Group calculus|Group calculus]]; [[Identity problem|Identity problem]]). Also, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007029.png" />, are arithmetic if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007030.png" />; see [[#References|[a3]]], [[#References|[a4]]] and [[Arithmetic group|Arithmetic group]].

There are several generalizations of Fibonacci groups, related to generalizations of Fibonacci numbers. D.L. Johnson [[#References|[a5]]] has introduced the generalized Fibonacci groups (see [[#References|[a9]]] for a survey)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007031.png" /></td> </tr></table>

where indices are taken modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007032.png" />. Another generalization of Fibonacci groups is due to C. Maclachlan [[#References|[a7]]] (see [[#References|[a8]]] for their geometrization):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007033.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007034.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007035.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f13007036.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.M. Campbell,   "Topics in the theory of groups" , ''Notes on Pure Math.'' , '''I''' , Pusan Nat. Univ.  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Conway,   "Advanced problem 5327"  ''Amer. Math. Monthly'' , '''72'''  (1965)  pp. 915</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Helling,   A.C. Kim,   J. Mennicke,   "A geometric study of Fibonacci groups"  ''J. Lie Theory'' , '''8'''  (1998)  pp. 1–23</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.L. Johnson,   "Extensions of Fibonacci groups"  ''Bull. London Math. Soc.'' , '''7'''  (1974)  pp. 101–104</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.C. Kim,   A. Vesnin,   "The fractional Fibonacci groups and manifolds"  ''Sib. Math. J.'' , '''38'''  (1997)  pp. 655–664</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C. Maclachlan,   A.W. Reid,   "Generalized Fibonacci manifolds"  ''Transformation Groups'' , '''2'''  (1997)  pp. 165–182</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  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</TD></TR></table>