Hyperbolic sections in surface bundles
Given a three manifold that fibers over the circle, the image of the circle by a section is a knot in the three manifold. The authors ask when its exterior is hyperbolic. For instance, when the three manifold is already hyperbolic, they show that the exterior of the section is always hyperbolic. For non hyperbolic bundles, they assume that the fiber is a surface \(F\) of genus at least 2. They view the three manifold as the mapping torus of an automorphism \(f\) of \(F\), called the monodromy. Namely, the bundle is the result of identifying the boundary components of \(F\times [0,1]\) via \(f\). The inverse image of the knot is an arc in \(F\times [0,1]\) that can be projected to the surface \(F\). In this paper the authors give a criterion of hyperbolicity using the projection of the arc to \(F\) and the monodromy \(f\). In fact they give two criteria, one when the manifold is Seifert fibered (\(f\) has finite order) and another one when the manifold has nontrivial JSJ splitting (\(f\) is reducible). This is a nice result, since both criteria correspond to the intuition of hyperbolicity: they say that the exterior of the section is hyperbolic provided that the projected arc in \(F\) is ``sufficiently complicated'', where this notion depends on the system of reducing curves in the second case.
scientific article; zbMATH DE number 5144527
scientific article