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The Fibonacci group $F(2,m)$ has the presentation (cf. also [[Finitely-presented group|Finitely-presented group]]; [[Presentation|Presentation]]):
 
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>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130070/f1300703.png" />.
+
where indices are taken modulo $m$.
  
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).
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Fibonacci groups were introduced by J.H. Conway [[#References|[a2]]] and are related to the [[Fibonacci numbers|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 [[#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]].
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Several combinatorial studies (see [[#References|[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 [[#References|[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|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|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]].
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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|Noetherian group]]). Since $M _ { 3 }$ is an affine [[Riemannian manifold|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|Hyperbolic group]]), the $F ( 2,2 n )$ are torsion-free, their Abelian subgroups are cyclic (cf. also [[Cyclic group|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|Group calculus]]; [[Identity problem|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 [[#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)
 
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>
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\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 <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):
+
where indices are taken modulo $m$. 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>
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\begin{equation*} F ^ { k } ( 2 , m ) = \end{equation*}
  
<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>
+
\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 [[#References|[a6]]] (which contains their geometrization as well):
 
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>
+
\begin{equation*} F ^ { k / l } ( 2 , m ) = \end{equation*}
  
<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>
+
\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====
 
====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>
+
<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>

Latest revision as of 16:46, 1 July 2020

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
How to Cite This Entry:
Fibonacci group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fibonacci_group&oldid=21713
This article was adapted from an original article by Boris N. Apanasov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article