Fibonacci manifolds as two-fold coverings of the three-dimensional sphere and the Meyerhoff-Neumann conjecture
type: Item
instance of: scholarly article
title: Fibonacci manifolds as two-fold coverings of the three-dimensional sphere and the Meyerhoff-Neumann conjecture
zbMATH Open document ID: 0882.57011
DOI: 10.1007/BF02104848
published in: Siberian Mathematical Journal
publication date: 1998-03-05T00:00:00Z
review text: 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 \(N\).
reviewed by: He Baihe
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 1023942
zbMATH Keywords: Fibonacci groups
Wikidata QID: Q122900756
author: Andrei Vesnin
author: Alexander Mednykh
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1023942