Holomorphic Removability of Julia Sets

Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.Comment: 48 pages. 9 PostScript figure

The good pants homology and the Ehrenpreis conjecture

We develop the notion of the good pants homology and show that it agrees with the standard homology on closed surfaces (the good pants are pairs of pants whose cuffs have the length nearly equal to some large number R). Combined with our previous work on the Surface Subgroup Theorem, this yields a proof of the Ehrenpreis conjecture.Comment: Revised to incorporate the advice of the referee. Appendix 2 has been substantially rewritten. 78 page

Nearly Fuchsian surface subgroups of finite covolume Kleinian groups

Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.Comment: v2: Final prepublication versio

Immersing almost geodesic surfaces in a closed hyperbolic three manifold

Let M be a closed hyperbolic three manifold. We construct closed surfaces which map by immersions into M so that for each one the corresponding mapping on the universal covering spaces is an embedding, or, in other words, the corresponding induced mapping on fundamental groups is an injection.Comment: One figure added and minor corrections. Probably the final versio

Counting essential surfaces in a closed hyperbolic three-manifold

Let M^3 be a closed hyperbolic three-manifold. We show that the number of genus g surface subgroups of π_1(M^3) grows like g^(2g)

A priori bounds for some infinitely renormalizable quadratics: II. Decorations

A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved is selected from a finite family of decorations. For this class of maps we prove {\it a priori} bounds. They imply local connectivity of the corresponding Julia sets and the Mandelbrot set at the corresponding parameter values.Comment: LaTeX, 29 pages, 2 figure