SURGERIES ON SMALL VOLUME HYPERBOLIC 3-ORBIFOLDS

Abstract

The computer program SnapPea by J. Weeks is a powerful tool for calculating the volumes of hyperbolic 3-manifolds. The small volume hyperbolic 3-manifolds have been studied rather intensively. The smallest known manifold M1, of volume 0.942707, was found independently by A. T. Fomenko and S. V. Matveev, and by J. Weeks. The ten smallest volume manifolds were described in [1, 2] where they were obtained in particular by Dehn surgeries on small volume hyperbolic knots and links. The structure of the set of volumes of hyperbolic 3-orbifolds is given in [3]. The computation of volumes of hyperbolic 3-orbifolds with singular sets obtainable by generalized surgeries on links in the three-dimensional sphere is possible due to SnapPea as well. If the singular set of a 3-orbifold is a graph other than a link, such a general tool is unavailable and the computation of volumes becomes a difficult problem that needs an individual approach in each particular case. It seems thus natural to study 3-orbifolds obtainable by surgeries on hyperbolic 3-orbifolds. The aim of this paper is to study closed hyperbolic 3-orbifolds obtainable by surgery on the smallest cusped hyperbolic 3-orbifolds and study coverings of these orbifolds by hyperbolic 3-manifolds obtainable by surgery on links. Recall [3, 4] that every closed hyperbolic 3-orbifold is obtainable by surgery on an orbifold with a nonrigid cusp (i.e., a cusp on which Dehn surgery, or Dehn filling, can be performed). Three smallest volume hyperbolic 3-orbifolds with nonrigid cusps were described by Adams [4]. Minimal regular coverings of these orbifolds are complements to the well-known links in the three-dimensional sphere. For example, the smallest orbifold with a nonrigid cusp is the Picard orbifold (the quotient of the hyperbolic three-dimensional space by the Picard group) which is covered by the complement of the Borromean rings; and the orbifolds obtainable by surgery on the Picard orbifold are covered by manifolds (generally, by cone-manifolds) obtainable by suitable surgeries on the Borromean rings. In this paper we establish an exact correspondence between the surgery parameters on the Adams orbifolds and their covering manifolds. This makes it possible to use the computer program SnapPea for calculating the volumes of hyperbolic 3-orbifolds. We only consider orientable 3-orbifolds, using the basic facts of the orbifold theory as in [3, 5]. Like the volumes of hyperbolic 3-manifolds, the volumes of hyperbolic 3-orbifolds form a well-ordered nondiscrete subset of order type ωω of the real axis R, and each volume is realized only for finitely many orbifolds [3]. In particular, there is a hyperbolic orbifold of smallest volume (which is not known yet), as well as the smallest limit volume. The singular set of the smallest known hyperbolic 3-orbifold is shown in Fig. 0.1 (the underlying space is the three-dimensional sphere; the edge labels 2 are omitted in the figure). Its volume is (approximately) 0.039050.