Fibonacci manifolds as two-fold coverings of the three-dimensional sphere and the Meyerhoff-Neumann conjecture
The Fibonacci groups \(F(2,m)\) were introduced by Conway and have the following presentation: \(F(2,m)= (x_1,x_2, \dots, x_m\); \(x_ix_{i+1} =x_{i+2}\), \(i\bmod m)\). Thus, the group \(F(2,2n)\), \(n\geq 2\), is realizable as a co-compact discrete group of isometries acting without fixed points on the space \(X_n\), where \(X_2=S^3\), \(X_3=E^3\), and \(X_n=H^3\) for \(n\geq 4\). A 3-manifold \(M_n= X_n/F(2,2n)\), \(n\geq 2\), uniformized by a Fibonacci group is referred to as a Fibonacci manifold. Denote by \(Th_n\), \(n\geq 2\), the closure of the 3-string braid \((\sigma_1 \sigma_2^{-1})^n\) given in canonical generators. One of the main results of the paper is the following Theorem 1. Each Fibonacci manifold \(M_n\), \(n\geq 2\), can be represented as a two-fold covering of \(S^3\) branched over the link \(Th_n\). Some closed orientable hyperbolic 3-manifold, denoted by \(N=W (3,-2;6,-1)\), was constructed by Dehn surgeries with parameters \((3,-2)\) and \((6,-1)\) on the components of the Whitehead link \(W\). The other result of the paper is the following Theorem 2. The Fibonacci manifold \(M_4\) is a two-fold unbranched covering of the Meyerhoff-Neumann manifold \(N\).
scientific article; zbMATH DE number 1023942
scientific article