Geometric limits and length bounds on curves
In the process of proving the ending lamination conjecture, Minsky showed the following theorem, which the author calls the length upper bound theorem. Let \(M\) be a hyperbolic 3-manifold with an isomorphism \(\phi: \pi_1(S) \rightarrow \pi_1(M)\) for some compact orientable surface \(S\) of hyperbolic type where each component of \(\partial S\) corresponds to a parabolic class of \(\pi_1(M)\). Then for any \(L\) there exists \(L'\) with the following property. Let \(\gamma\) and \(\delta\) be two essential simple closed curves such that the closed geodesics representing the free homotopy classes of \(\phi(\gamma)\) and \(\phi(\delta)\) have lengths less than \(L\). Then for any vertex \(c\) appearing in a tight geodesic in the curve complex of \(S\) connecting \(\gamma\) with \(\delta\), the closed geodesic in \(M\) representing the free homotopy class of \(\phi(c)\) has length less than \(L'\). In this paper, a new proof of this theorem is given. The original proof of this theorem by Minsky relies on the machinery of hierarchies of tight geodesics which he developed in joint work with Masur. Bowditch gave an alternative proof which uses hyperbolic 3-manifolds more and dispenses with hierarchies. The author's approach in the paper under review is closer to the one by Bowditch, but uses geometric limits even for the cases of once-punctured torus groups and four-times punctured sphere groups, where Bowditch uses an argument based on direct computation of traces.
scientific article; zbMATH DE number 5950741
scientific article