Hyperbolic volumes of Fibonacci manifolds
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.
scientific article; zbMATH DE number 923497
scientific article