Departmental Colloquium
Jeffrey Brock
Brown University
Length spectra of moduli spaces and volumes of hyperbolic surface bundles
Abstract: Given a closed surface S of negative Euler-characteristic, a self-homeomorphism of S that is not isotopic to the identity is called "pseudo-Anosov" if it preserves no family of simple closed curves on S up to isotopy. Due to Thurston, the notion of a pseudo-Anosov homeomorphism plays a dual role in Teichmüller theory, where it induces a hyperbolic isometry of the Weil-Petersson metric on Teichmüller space, and in the study of 3-dimensional manifolds, where it serves as the monodromy for a surface bundle over the circle with a complete hyperbolic structure.
There is a fixed constant K depending only on the Euler characteristic of S so that for a given pseudo-Anosov f:S->S the volume of the corresponding surface bundle M_f and the translation length of f on Teichmüller space have ratio lying in the interval [1/K,K]. This relationship motivates questions on nature and depth of the connection between these two invariants of the homeomorphism f.
We suggest and develop some new connections between the collection of volumes of surface bundles and the Weil-Petersson translation distances of their pseudo-Anosov monodromy maps, while introducing a fundamental confounding disconnect between these two quantities in general, answering a question of Manin and Marcolli.
Friday February 26, 2010 at 3:00 PM in SEO 636