TAILIEUCHUNG - Đề tài "Universal bounds for hyperbolic Dehn surgery"

This paper gives a quantitative version of Thurston’s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic conemanifold structures, using infinitesimal harmonic deformations and analysis of geometric limits. | Annals of Mathematics Universal bounds for hyperbolic Dehn surgery By Craig D. Hodgson and Steven P. Kerckhoff Annals of Mathematics 162 2005 367 421 Universal bounds for hyperbolic Dehn surgery By Craig D. Hodgson and Steven P. Kerckhoff Abstract This paper gives a quantitative version of Thurston s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3-manifold and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic conemanifold structures using infinitesimal harmonic deformations and analysis of geometric limits. 1. Introduction If X is a noncompact finite volume orientable hyperbolic 3-manifold it is the interior of a compact 3-manifold with a finite number of torus boundary components. For each torus there are an infinite number of topologically distinct ways to attach a solid torus. Such Dehn fillings are parametrized by pairs of relatively prime integers once a basis for the fundamental group of the torus is chosen. If each torus is filled the resulting manifold is closed. A fundamental theorem of Thurston 43 states that for all but a finite number of Dehn surgeries on each boundary component the resulting closed 3-manifold has a hyperbolic structure. However it was unknown whether or not the number of such nonhyperbolic surgeries was bounded independent of the original noncompact hyperbolic manifold. In this paper we obtain a universal upper bound on the number of nonhy-perbolic Dehn surgeries per boundary torus independent of the manifold X . There are at most 60 nonhyperbolic Dehn surgeries if there is only one cusp if there are multiple cusps at most 114 surgery curves must be excluded from each boundary torus. The research of the first author was partially supported by grants from the ARC. The research of the second author was partially supported by grants .

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