Surgeries on small volume hyperbolic 3-orbifolds. (Russian, English) Zbl 0977.57016

Vesnin, A. Yu.; Mednykh, A. D.; Zimmermann, B.

Sib. Mat. Zh. 42, No. 2, 1035-1051 (2001); translation in Sib. Math. J. 42, No. 2, 271-281 (2001).

The goal is to study closed hyperbolic 3-orbifolds, obtained by the Dehn surgery on the cusps of noncompact hyperbolic 3-orbifolds of the smallest volumes, and the coverings of the orbifolds by hyperbolic 3-manifolds, obtained by the Dehn surgery on links. A connection with the hyperbolic 3-manifolds makes it possible to calculate the volumes of the orbifolds and their coverings using Jeff Weeks’ SnapPea program.
The attention is focused on the three smallest hyperbolic 3-orbifolds with nonrigid cusp. These orbifolds were obtained by Colin C. Adams in [Mich. Math. J. 46, No. 3, 515-531 (1999; Zbl 0961.57010)]. The first of them is the well-known Picard orbifold. For these three orbifolds and their covering manifolds, the coverings diagrams are studied in detail. A connection between the parameters of the surgeries on the orbifolds and the corresponding manifolds is established and applied for calculating the volumes.

Reviewer: M.V.Belolipetsky (Novosibirsk)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57M10	Covering spaces and low-dimensional topology

Keywords:

hyperbolic manifold; orbifold; volume of a hyperbolic orbifold; Dehn surgery

Citations:

Zbl 0961.57010