×

Rigidity of cusps in deformations of hyperbolic 3-orbifolds. (English) Zbl 0813.57013

Starting with a 3-manifold with a complete hyperbolic structure of finite volume, a theory originated by Thurston allows one to consider deformations of this structure to manifolds which are not complete, but whose completions are the result of filling in some of the cusps with solid tori. The topological type of the manifold is changed, and in place of the cusp is a geodesic which arises from the core circle of the solid torus. This process, known as Dehn filling, can also be adapted to hyperbolic 3-orbifolds. A natural question is how the Euclidean structure associated to the other cusps changes during such deformations.
In previous work [Math. Proc. Camb. Philos. Soc. 109, No. 3, 509-515 (1991; Zbl 0728.57009)] the authors constructed examples of 2-cusped hyperbolic 3-orbifolds with the property that deforming one of the cusps by any hyperbolic Dehn filling while keeping the other cusp complete left the Euclidean structure of the complete cusp unaffected. This unaffected cusp is said to be geometrically isolated from the other one. In the paper under review, the authors give a number of refinements of this construction. In particular, they prove that there are infinitely many 2- cusped hyperbolic 3-manifolds whose cusps are geometrically isolated from each other, thereby disproving a conjecture of another author. They also construct such an example which is arithmetic. A 2-cusped example is constructed which has one cusp geometrically isolated from the other, yet the other is not geometrically isolated from the first. Refinements of the concept of geometric isolation are introduced, examined, and related to the analytic function \(\Phi\) on the character variety, introduced by the first author and D. Zagier. The paper is very well-written, with useful figures and a number of explicit volume calculations obtained using Snappea.

MSC:

57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
57N10 Topology of general \(3\)-manifolds (MSC2010)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Citations:

Zbl 0728.57009
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] [CL] Cooper, D., Long, D.D.: An undetected slope in a knot manifold. In: Topology ’90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University, pp. 111-121. Berlin New York: de Gruyter 1992 · Zbl 0771.57006
[2] [EP] Epstein, D.B.A., Penner, R.C.: Euclidean decompositions of non-compact hyperbolic manifolds. J. Differ. Geom.27, 67-80 (1988) · Zbl 0611.53036
[3] [HMW] Hodgson, C.D., Meyerhoff, G.R., Weeks, J.R.: Surgeries on the Whitehead link yield geometrically similar manifolds. In: Topology ’90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University, pp. 195-206. Berlin New York. de Gruyter 1992 · Zbl 0767.57007
[4] [K1] Kapovich, M.: On faithful matrix representations of fundamental groups of 3-manifolds with toroidal boundary. (Preprint)
[5] [K2] Kapovich, M.: Eisenstein series and Dehn surgery. (Preprint)
[6] [KM] Kojima, S., Miyamoto, Y.: The minimum volume of hyperbolic 3-manifolds with totally geodesic boundary. (Preprint) · Zbl 0729.53042
[7] [NR1] Neumann, W.D., Reid, A.W.: Arithmetic of hyperbolic 3-manifolds. In: Topology ’90, Proceedings of the Research Semester in Low Dimensional Topology at Ohio State University, pp. 273-310. Berlin New York: de Gruyter, 1992
[8] [NR2] Neumann, W.D., Reid, A.W.: Amalgamation and the invariant trace-field of Kleinian groups. Math. Proc. Camb. Philos. Soc.109, 509-515 (1991) · Zbl 0728.57009 · doi:10.1017/S0305004100069942
[9] [NZ] Neumann, W.D., Zagier, D.: Volumes of hyperbolic 3-manifolds. Topology24, 307-332 (1985) · Zbl 0589.57015 · doi:10.1016/0040-9383(85)90004-7
[10] [R] Reid, A.W.: Ph. D. Thesis. University of Aberdeen (1987)
[11] [T] Thurston, W.P.: The geometry and topology of 3-manifolds. Mimeographed lecture notes, Princeton University (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.