Three-dimensional hyperbolic manifolds of small volume with three hyperelliptic involutions
A three-dimensional manifold \(M^3\) is called hyperelliptic if it admits an involution \(\tau\) such that \(M^3/\langle\tau\rangle\) is homeomorphic to the sphere \(S^3\). In this case, \(\tau\) is called a hyperelliptic involution. Existence of hyperelliptic manifolds in each of the eight three-dimensional Thurston geometries was proven by \textit{A.~D. Mednykh} [Ann. Global Anal. Geom. 8, No. 1, 13-19 (1990; Zbl 0712.57007)]. In the article under review, hyperelliptic involutions on hyperbolic manifolds of small volume are studied. Actually, hyperbolic 3-manifolds of smallest volumes were not known by now, but there exists a list of ten hyperbolic 3-manifolds of smallest known volumes which was obtained using the computer program SnapPea and described by \textit{C.~D. Hodgson} and \textit{J.~R. Weeks} [Exp. Math. 3, No. 4, 261-274 (1994; Zbl 0841.57020)]. As is shown by \textit{A.~Yu. Vesnin} and \textit{A.~D. Mednykh} [Sib. Math. J. 37, No. 5, 893-897 (1996); translation from Sib. Mat. Zh. 37, No. 5, 1013-1018 (1996; Zbl 0887.57021)], these manifolds may be obtained by a Dehn surgery \(W(m/n, p/q)\) on the two components of the Whitehead link \(W\). This fact allows us to provide a uniform approach to studying these manifolds. The particular interests of the article under review are hyperelliptic involutions of \(W(m/n; p/q)\) and factor orbifolds obtained by the actions of involutions.
scientific article; zbMATH DE number 1396473