Exceptional Dehn fillings on hyperbolic 3-manifolds with at least two boundary components
The author deals with exceptional Dehn fillings. Let \(M\) be a compact, orientable, connected 3-manifold with a torus boundary component \(\partial_0 M\) and denote by \(M(r)\) the 3-manifold obtained from \(M\) by Dehn filling on the component \(\partial_0 M\) along the slope \(r\). It is a well known result of \textit{W. Thurston} [Bull. Am. Math. Soc. 6, 357--381 (1982; Zbl 0496.57005)] that if \(M\) is hyperbolic then \(M(r)\) will be hyperbolic for all but a finite number of slopes called exceptional. The understanding of exceptional slopes is a widely studied problem. In this article the author analyzes the case of a hyperbolic 3-manifold \(M\) with a torus boundary component and at least one other boundary component. He proves that 6 is an upper bound for the number of exceptional slopes and that any two exceptional slopes have mutual distance (i.e. minimal geometric intersection number) no larger than 4, unless \(M\) is the exterior of the Whitehead sister link, in which case there are two exceptional slopes at distance 5. Since, by Thurston's Geometrization Theorem for Haken manifolds [\textit{W. Thurston}, Bull. Am. Math. Soc. 6, 357--381 (1982; Zbl 0496.57005)], if \(r\) is an exceptional slope then \(M(r)\) contains an essential torus or Klein bottle, the result is achieved by studying the relationship between the distance of two different exceptional slopes \(r_1,r_2\) and the essential surfaces (tori or Klein bottles) contained in \(M(r_1),M(r_2)\) and by applying known results of Gordon, Luecke, Oh, Qiu, Scharlemann and Wu [see \textit{C. McA. Gordon}, in Proceedings of the Kirbyfest, Berkeley, CA, USA, June 22-26, 1998, 177--199 (1999; Zbl 0948.57014)].
scientific article; zbMATH DE number 5178556
scientific article