The study of hyperbolic 3-manifolds draws together the beautiful subjects of discrete group actions, conformal dynamical systems, and hyperbolic geometry. Thurston’s geometrization conjecture and his work on the Haken case led to a beautiful new geometric perspective on the deformation space raising the question of how parameters “at infinity” determine the interior geometry of the manifold. The Model Manifold Theorem of [BCM] establishes a combinatorial model for the ends of such a manifold, proving Thurston’s ending lamination conjecture. Ongoing work seeks to make these models effective, and to relate them to new geometric flows on the deformation space connecting the deformation theory to Witten’s notion of Renormalized Volume from conformal field theory.
(With Dick Canary and Yair Minsky). Annals of Math 176 (2012), pp. 1-149.
(With Ken Bromberg). Math. Res. Lett. 23 (2016) pp. 649-674.