Reducing and toroidal Dehn fillings on 3-manifolds bounded by two tori (Q867363)

The aim of this paper is to find bounds on the geometric intersection number between exceptional slopes on hyperbolic manifolds with toral boundary. A slope is the isotopy class of an unoriented simple closed curve on the 2-torus. A slope determines a Dehn filling, i.e. the result of gluing a solid torus by adding first a 2-handle along the slope and then a 3-handle. The slope is called reducible, toroidal or exceptional if the corresponding filled manifold is reducible, toroidal or non-hyperbolic. The study of intersection numbers between those slopes has been crucial in the proof of results as the cyclic surgery theorem [\textit{M. Culler, C. McA. Gordon, J. Luecke} and \textit{P. B. Shalen}, Ann. Math. (2) 125, 237--300 (1987; Zbl 0633.57006)]. Here is the main theorem of the paper. Let \(M\) be a hyperbolic manifold with two boundary components, which are tori. Looking at Dehn fillings on only one of the components, the geometric intersection number between a reducible and a toroidal slope is at most two. An example of \textit{M. Eudave-Muñoz} and \textit{Y.-Q. Wu} [Pac. J. Math. 190, No. 2, 261--275 (1999; Zbl 1011.57005)] shows that this bound is sharp. This answers a question raised by \textit{C. McA. Gordon} in [Proceedings of the Kirbyfest (Berkeley, CA, 1998), 177--199 (electronic), Geom. Topol. Publ., Coventry (1999; Zbl 0948.57014)], where the interested reader will find a complete survey of such kind of bounds on geometric intersection numbers. Similar bounds are obtained in this paper, all of them proved with combinatorial techniques, including Scharlemann cycles.