View source for Fibonacci manifold

{{TEX|done}}
The Fibonacci manifold $M_n$, $n\geq2$, is a closed orientable [[Three-dimensional manifold|three-dimensional manifold]] whose fundamental group is the [[Fibonacci group|Fibonacci group]] $F(2,2n)$ (cf. also [[Orientation|Orientation]]). Such manifolds were discovered by H. Helling, A.C. Kim and J. Mennicke [[#References|[a2]]] as geometrizations of Fibonacci groups. For $n\geq4$, the manifolds $M_n$ are closed hyperbolic three-manifolds (cf. also [[Hyperbolic metric|Hyperbolic metric]]), $M_3$ is the Euclidean Hantzche–Wendt manifold, and $M_2$ is the [[Lens space|lens space]] $L(5,2)$ (see [[#References|[a2]]]).

Many interesting properties of Fibonacci manifolds can be obtained from their relation to links and branched coverings over the three-dimensional sphere $S^3$, discovered by H.M. Hilden, M.T. Lozano and J.M. Montesinos [[#References|[a3]]]. In fact,

1) $M_n$ is the $n$-fold cyclic covering of the three-dimensional sphere $S^3$, branched over the figure-eight knot (cf. [[Listing knot|Listing knot]]), see [[#References|[a3]]];

2) $M_n$ can be obtained by [[Dehn surgery|Dehn surgery]] with parameters $1$ and $-1$ on the components of the chain of $2n$ linked circles in $S^3$, see [[#References|[a1]]];

3) $M_n$ is the two-fold covering of $S^3$, branched over the link $T_n$ corresponding to the closed $3$-string braid $(\sigma_1\sigma_2^{-1})^n$, see [[#References|[a9]]]. The above well-known family $T_n$ of links in $S^3$ includes the figure-eight knot as $T_2$, the Borromean rings as $T_3$, the Turk's head knot $8_{18}$ as $T_4$, and the knot $10_{123}$ as $T_5$ (in the notation of [[#References|[a7]]]). The last description of $M_n$ also shows that the hyperbolic volumes of the compact Fibonacci manifolds $M_{2n}$, $n\geq2$, coincide with those ones of the (non-compact) link complements $S^3\setminus T_n$, see [[#References|[a8]]], [[#References|[a9]]]. Also, since the $M_n$ are arithmetic if and only if $n=4,5,6,8,12$ (see [[#References|[a2]]], [[#References|[a3]]] and [[Arithmetic group|Arithmetic group]]), this shows that hyperbolic manifolds with the same volume can be both arithmetic and non-arithmetic, see [[#References|[a8]]].

There are several generalizations of Fibonacci manifolds, related to generalizations of the Fibonacci groups, see [[#References|[a10]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]] and [[Fibonacci group|Fibonacci group]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Cavicchioli,   F. Spaggiari,   "The classification of 3-manifolds with spines related to Fibonacci groups" , ''Algebraic Topology, Homotopy and Group Cohomology'' , ''Lecture Notes in Mathematics'' , '''1509''' , Springer  (1992)  pp. 50–78</TD></TR><TR><TD valign="top">[a2]</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">[a3]</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">[a4]</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">[a5]</TD> <TD valign="top">  C. Maclachlan,   "Generalizations of Fibonacci numbers, groups and manifolds"  A.J. Duncan (ed.)  N.D. Gilbert (ed.)  J. Howie (ed.) , ''Combinatorial and Geometric Group Theory (Edinburgh, 1993)'' , ''Lecture Notes'' , '''204''' , London Math. Soc.  (1995)  pp. 233–238</TD></TR><TR><TD valign="top">[a6]</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">[a7]</TD> <TD valign="top">  D. Rolfson,   "Knots and links" , Publish or Perish  (1976)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  A.Yu. Vesnin,   A.D. Mednykh,   "Hyperbolic volumes of Fibonacci manifolds"  ''Sib. Math. J.'' , '''36''' :  2  (1995)  pp. 235–245</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  A.Yu. Vesnin,   A.D. Mednykh,   "Fibonacci manifolds as two-fold coverings over the three-dimensional sphere and the Meyerhoff–Neumann conjecture"  ''Sib. Math. J.'' , '''37''' :  3  (1996)  pp. 461–467</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  B.N. Apanasov,   "Conformal geometry of discrete groups and manifolds" , de Gruyter  (2000)</TD></TR></table>