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On hyperbolic 3-manifolds realizing the maximal distance between toroidal Dehn fillings. (English) Zbl 1082.57011

Let \(M\) be a hyperbolic \(3\)-manifold with a torus boundary component \(T_0\), in the sense that \(M\) with its boundary tori removed admits a complete hyperbolic structure with totally geodesic boundary. For a slope \(\gamma\) on \(T_0\), \(\gamma\)-Dehn filling \(M(\gamma)\) is hyperbolic except finitely many slopes, which are referred to as exceptional slopes. The case where \(M(\gamma)\) contains an essential torus is typical, and such a slope is said to be toroidal. The paper under review focuses on this situation. For two slopes, the distance between them is the minimal geometric intersection number. Roughly speaking, an upper bound for the distance between exceptional slopes gives that for the number of exceptional slopes.
C. Gordon [Trans. Am. Math. Soc. 350, 1713–1790 (1998; Zbl 0896.57011)] showed that if two slopes \(\gamma_1\) and \(\gamma_2\) are toroidal then the distance \(\Delta\) between them is at most eight, and there are only four specific manifolds with \(\Delta\geq 6\). A manifold \(M\) is said to be large if \(H_2(M,\partial M-T_0)\not=0\), following Y. Q. Wu [J. Differ. Geom. 48, 407–437 (1998; Zbl 0917.57015)]. Since the above four specific manifolds are not large, \(\Delta\leq 5\) for a large \(M\) manifold. C. McA. Gordon [Geom. Topol. Monogr. 2, 177–199 (1999; Zbl 0948.57014)] asked if there is a large \(M\) with \(\Delta=5\).
The main result of the paper under review is that \(\Delta\leq 4\), unless the boundary of \(M\) consists of at most two tori. Thus a broad class of large manifolds cannot admit two toroidal slopes at distance five. This is also the first step to determine all hyperbolic \(3\)-manifolds that admit two toroidal slopes at distance five. The proof is based on a long analysis of the graph pair coming from the intersection of two punctured tori as in earlier works in this area.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57M25 Knots and links in the \(3\)-sphere (MSC2010)
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