Harmonic maps between 3-dimensional hyperbolic spaces

We prove that a quasiconformal map of the 2-sphere admits a harmonic quasi-isometric extension to the 3-dimensional hyperbolic space, thus confirming the well known Schoen Conjecture in dimension 3.Comment: Final Versio

Criterion for Cannon's Conjecture

The Cannon Conjecture from the geometric group theory asserts that a word hyperbolic group that acts effectively on its boundary, and whose boundary is homeomorphic to the 2-sphere, is isomorphic to a Kleinian group. We prove the following Criterion for Cannon's Conjecture: A hyperbolic group $G$ (that acts effectively on its boundary) whose boundary is homeomorphic to the 2-sphere is isomorphic to a Kleinian group if and only if every two points in the boundary of $G$ are separated by a quasi-convex surface subgroup. Thus, the Cannon's conjecture is reduced to showing that such a group contains "enough" quasi-convex surface subgroups.Comment: Revised versio

Harmonic diffeomorphisms of noncompact surfaces and Teichmüller spaces

Let g : M -> N be a quasiconformal harmonic diffeomorphism between noncompact Riemann surfaces M and N. In this paper we study the relation between the map g and the complex structures given on M and N. In the case when M and N are of finite analytic type we derive a precise estimate which relates the map g and the Teichmüller distance between complex structures given on M and N. As a corollary we derive a result that every two quasiconformally related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. In addition, we study the question of whether every quasisymmetric selfmap of the unit circle has a quasiconformal harmonic extension to the unit disk. We give a partial answer to this problem. We show the existence of the harmonic quasiconformal extensions for a large class of quasisymmetric maps. In particular it is proved that all symmetric selfmaps of the unit circle have a unique quasiconformal harmonic extension to the unit disk

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

Non-realizability of the Torelli group as area-preserving homeomorphisms

Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion-free subgroups of $\text{Mod}(S_g)$. The main result of this paper is that the Torelli group has no realization inside the area-preserving homeomorphisms.Comment: 22 pages, 5 figure

Homology of curves and surfaces in closed hyperbolic 3-manifolds

Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed $\pi_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, $\pi_1$-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented compact $\pi_1$-injective immersed quasi-Fuchsian subsurface. In part, we exploit techniques developed earlier by Kahn and Markovic about good pants constructions, but we only distill geometric and topological ingredients from their papers so no hard analysis is involved in this paper.Comment: 66 pages, 8 figures, with additional explanations and minor correction

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

The mapping class group cannot be realized by homeomorphisms

Let $M$ be a closed surface. By \Homeo(M) we denote the group of orientation preserving homeomorphisms of $M$ and let \MC(M) denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says that for any closed surface $M$ of genus \g \ge 2, there is no homomorphic section \E:\MC(M) \to \Homeo(M) of the standard projection map \Proj:\Homeo(M) \to \MC(M).Comment: 33 pages, 6 figure

The Moduli space of Riemann Surfaces of Large Genus

Let $\mathcal{M}_{g,\epsilon}$ be the $\epsilon$-thick part of the moduli space $\mathcal{M}_g$ of closed genus $g$ surfaces. In this article, we show that the number of balls of radius $r$ needed to cover $\mathcal{M}_{g,\epsilon}$ is bounded below by $(c_1g)^{2g}$ and bounded above by $(c_2g)^{2g}$, where the constants $c_1,c_2$ depend only on $\epsilon$ and $r$, and in particular not on $g$. Using the counting result we prove that there are Riemann surfaces of arbitrarily large injectivity radius that are not close (in the Teichm\"uller metric) to a finite cover of a fixed closed Riemann surface. This result illustrates the sharpness of the Ehrenpreis conjecture.Comment: v2, accepted in GAFA, updates based on referee's comment