Skinning maps

Let M be a hyperbolic 3-manifold with nonempty totally geodesic boundary. We prove that there are upper and lower bounds on the diameter of the skinning map of M that depend only on the volume of the hyperbolic structure with totally geodesic boundary, answering a question of Y. Minsky. This is proven via a filling theorem, which states that as one performs higher and higher Dehn fillings, the skinning maps converge uniformly on all of Teichmuller space. We also exhibit manifolds with totally geodesic boundaries whose skinning maps have diameter tending to infinity, as well as manifolds whose skinning maps have diameter tending to zero (the latter are due to K. Bromberg and the author). In the final section, we give a proof of Thurston's Bounded Image Theorem.Comment: 50 pages, 4 figures. v3. Major revision incorporating referees' comments. To appear in the Duke Mathematical Journal. v2. Cosmetic changes, minor corrections, inclusion of theorem with K. Bromber

Achievable ranks of intersections of finitely generated free groups

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups.Comment: 4 pages, 4 figure

Slicing, skinning, and grafting

We prove that a Bers slice is never algebraic, meaning that its Zariski closure in the character variety has strictly larger dimension. A corollary is that skinning maps are never constant. The proof uses grafting and the theory of complex projective structures.Comment: 11 pages, 1 figure, to appear in American Journal of Mathematic

A geometric criterion to be pseudo-Anosov

We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface, and we show that such a class is pseudo-Anosov if its geodesic representative is "wide" in some hyperbolic 3-manifold homeomorphic to the trivial interval bundle over S.Comment: v2. To appear in the Michigan Mathematical Journal. Revised according to referees' comments. 24 pages, no figures. v1. 21 pages, no figure