Quasi-conformal extensions of quasi-isometries of hyperbolic space

This talk will motivate and describe (what Marc Bourdon calls) a "Morse Lemma" which asserts that every quasi-geodesic in hyperbolic space lies at bounded distance from an actual geodesic. Following Bourdon's "Quasi-conformal geometry and Mostow rigidity", we will then use this fact to show that every quasi-isometry of hyperbolic space has a quasi-conformal extension to the boundary at infinity. This is a key ingredient in the proof of Mostow rigidity using quasi-conformal geometry.