Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance
A Dehn filling on a boundary component of a three manifold that is homeomorphic to a 2-torus is the result of gluing a solid torus \(D^2\times S^1\), thus it is a manifold with one boundary component less. Such a filling is determined by a slope on the boundary torus \(T^2\), namely the homology class of the meridian curve \(D^2\times *\) in \(H_1(T^2)\), up to sign. By Thurston's hyperbolic Dehn filling, all but finitely many Dehn fillings on a given component are hyperbolic. The remaining fillings are called exceptional and have been intensively studied. For instance, toroidal fillings are exceptional, because a toroidal manifold contains an essential torus, hence it cannot be hyperbolic. One can define a distance between Dehn fillings by means of the absolute value of the intersection number of their meridian slopes in \(T^2\). The main result of this paper asserts that if the distance between two toroidal slopes is at least 3, then the boundary of the initial three manifold contains at most three tori \(T^2\). This answers a question of \textit{C. McA. Gordon} [in: Hass, Joel (ed.) et al., Proceedings of the Kirbyfest, Berkeley, CA, USA, June 22--26, 1998. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 2, 177--199 (1999; Zbl 0948.57014)]. As a consequence, the authors obtain an optimal bound for the number of exceptional slopes on a hyperbolic 3-manifold whose boundary contains at least 4 tori. The proof uses the intersection diagrams on the essential torus of the toroidal Dehn fillings, and analyzes the Scharlemann cycles, among other things. The authors prove it first for the case that the manifolds contain Klein bottles. The case where the distance of the slopes is at least 4 has been proved by the authors and Gordon-Wu in previous works.
scientific article; zbMATH DE number 5550610
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