Hyperbolic volumes of Fibonacci manifolds. (English. Russian original) Zbl 0865.57012

Vesnin, A. Yu.; Mednykh, A. D.

Sib. Math. J. 36, No. 2, 235-245 (1995); translation from Sib. Mat. Zh. 36, No. 2, 266-277 (1995).

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 H. Helling, A. C. Kim and J. Mennicke (preprint 343, Univ. of Bielefeld) and by H. M. Hilden, M. T. Lozano and 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.

Reviewer: B.Apanasov (Norman)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57M05	Fundamental group, presentations, free differential calculus
20F05	Generators, relations, and presentations of groups

Keywords:

Fibonacci manifolds; Fibonacci groups; hyperbolic 3-manifolds

Citations:

Zbl 0767.57004

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