An upper bound for injectivity radii in convex cores (Q1943730)

The paper under review proves that, for a hyperbolic three manifold \(N\) with finitely generated fundamental group, there is an upper bound on the radius of a hyperbolic ball embedded in the convex core of \(N\), that depends only on the topology of \(N\). Partial cases of this theorem had been proved before by Canary for product manifolds, by Fan for books or \(I\)-bundles and acylindrical manifolds, and by Evans in the incompressible case. Using the characteristic submanifold theorem, for a given homotopy type there are finitely many possibilities for the topology, thus one can assume that the bound depends only on the fundamental group of \(N\). In fact the following result is proved in the paper: The convex hull of \(N\) is contained in the union of the \(t\)-thin part of \(N\) and a union of \(\xi\)-handles, where \(t>0\) depends on the number of simplices of a triangulation of \(N\) and a given \(\xi>0\). Here the \(\xi\)-handles are 2-handles attached to the thin part with core discs of diameter and circumference at most \(\xi\). As the author notices, this set of 2-handles can be thought, up to bounded Hausdorff distance, as the set of points in the convex core contained in some compressing disc of diameter and circumference at most \(\xi\). The proof involves constructing sequences of homotopies and compressions of singular hyperbolic surfaces, and before the general case, the convex cocompact case and the case without cusps are considered. As an application, it is proved that if a sequence of Kleinian groups converges algebraically and geometrically, then the limit sets converge, for the Hausdorff topology. The proof is rather straightforward from the main theorem of the paper and a result of [\textit{C. T. McMullen}, Renormalization and 3-manifolds which fiber over the circle. Annals of Mathematics Studies. 142. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0860.58002)].