Hyperbolic volumes of Fibonacci manifolds
type: Item
instance of: scholarly article
title: Hyperbolic volumes of Fibonacci manifolds
zbMATH Open document ID: 0865.57012
DOI: 10.1007/BF02110146
author: Andrei Vesnin
author: Alexander Mednykh
published in: Siberian Mathematical Journal
publication date: 1997-06-15T00:00:00Z
review text: The authors study algebra, topology and arithmetics of the Fibonacci manifolds, that is closed hyperbolic 3-manifolds whose fundamental groups are the Fibonacci groups \[ F(2,m)=\langle x_1,\dots ,x_m ; x_ix_{i+1}=x_{i+2} , i\text{ mod }m\rangle. \] In fact, due to celebrated results by \textit{H. Helling}, \textit{A. C. Kim} and \textit{J. Mennicke} (preprint \textbf{343}, Univ. of Bielefeld) and by \textit{H. M. Hilden}, \textit{M. T. Lozano} and \textit{J. M. Montesinos-Amilibia} [Topology '90, Contrib. Res. Semester Low Dimensional Topol., Columbus/OH (USA) 1990, Ohio State Univ. Math. Res. Inst. Publ. 1, 169-183 (1992; Zbl 0767.57004)], such hyperbolic 3-manifolds correspond to the Fibonacci groups \(F(2,2n)\) with \(n\geq 4\) and are \(n\)-fold cyclic coverings of the 3-sphere branched over the figure-eight knot. In particular, the authors show that the hyperbolic volumes of the (closed hyperbolic) Fibonacci manifolds coincide with volumes of non-compact hyperbolic 3-manifolds (some known knot and link complements). As a corollary of this fact, the authors present arithmetic and non-arithmetic hyperbolic 3-manifolds with the same volume.
reviewed by: Boris N. Apanasov
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 923497
zbMATH Keywords: Fibonacci manifolds
zbMATH Keywords: Fibonacci groups
zbMATH Keywords: hyperbolic 3-manifolds
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 923497