Appears in collection : 2022 - T2 - WS1 - Mapping class groups and Out(Fn)

Thurston’s proof of the hyperbolization theorem for Haken manifolds involved a gluing step, in which the matching conditions for the boundary components being glued are phrased in terms of a fixed-point problem for a certain self-map of Teichmuller space. A better quantitative understanding of this process would improve our control of the relation of topology to geometry of these manifolds. Thurston stated an appealing theorem: that a finite power of the self-map has bounded image, thus controlling the process of finding the fixed point. Nobody seems to know what Thurston’s proof was. With Ken Bromberg and Dick Canary, we provide a proof that involves building uniform models for the internal geometry of hyperbolic manifolds of a given topological type. I will try to explain the background and the ingredients of this theorem, which include the machinery of hierarchical structure in the mapping class group.