Length Bounds in Quasifuchsian Manifolds

by Kenneth Bromberg (University of Utah)

Monday Jun 16, 2025, 4:00 PM → 5:15 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

A quasifuchsian manifold is a hyperbolic structure on the product of a surface and the line that is naturally compactified by two conformal structures at infinity. By a classical result of Bers, curves that have bounded length in the hyperbolic structures on these surfaces also have bounded length in the hyperbolic 3-manifold. However, the converse fails — one can construct examples of quasifuchsian manifolds that contain curves of bounded length in the 3-manifold while the curves are arbitrarily long in the hyperbolic structures at infinity. To rectify this, Minsky gave a description of the bounded length curves in the 3-manifold in terms of the data at infinity using the curve complex. These a priori bounds played a central role in the Brock-Canary-Minsky proof of the ending lamination conjecture. Bowditch later gave a new proof of these bounds. We will describe another proof of this result. While it uses many of the ideas of the approaches of Minsky and Bowditch, unlike their proofs the result is effective.