Isometries of hyperbolic Fibonacci manifolds
A Fibonacci manifold \(M_n\) is a compact orientable 3-manifold whose fundamental group is the Fibonacci group \(F(2,2n)\). For \(n\geq 4\), each of these manifolds has hyperbolic structure. In the article under review, the group \(\text{Isom}(M_n)\) of isometries of a hyperbolic Fibonacci manifold is studied. The main result is a complete description of \(\text{Isom}(M_n)\) for \(n\geq 6\). The authors prove that, for \(n\geq 6\), the group \(\text{Isom}(M_n)\) consists of \(8n\) elements and has a representation \(\langle x, y \mid x^{2n}=y^4=(yx)^2=(y^{-1}x)^2=1 \rangle .\) This result is obtained by geometric methods and significantly uses a precise construction of the fundamental set for a 2-bridge link orbifold introduced by \textit{A. D. Mednykh} and \textit{A. A. Rasskazov} [Univ. of Bielefeld, Preprint 98-062 (1998)]. In the arithmetic case \((n=4,5,6,8,12)\), the isometry group of a Fibonacci manifold was previously studied by \textit{C. Maclachlan} and \textit{A. W. Reid} [Transform. Groups 2, No. 2, 165-182 (1997; Zbl 0890.57023)].
scientific article; zbMATH DE number 1285557
scientific article