May 7, 2014 THE SURFACE SUBGROUP AND THE EHRENEPREIS CONJECTURES JEREMY KAHN AND VLADIMIR MARKOVIC Abstract. We survey our recent results including the Surface Subgroup Theorem and the Ehrenpreis Conjecture. Applications and future direction are discussed. 1. Introduction 1.1. The Surface Subgroup Theorem. One of the corollaries of the Geometrization Theorem is that most 3-manifolds admit hyperbolic structure. Therefore when studying topology of a 3-manifold one can often assume that the manifold is endowed with a hyperbolic structure. This greatly expands the tool-kit that is available bringing hyperbolic geometry, analysis and dynamics into play. An essential step in the eventual proof of the Virtual Haken and the Virtual Fibering Conjectures is the Surface Subgroup Theorem: Theorem 1.1 (Kahn-Markovic). Every closed hyperbolic 3-manifold contains a quasifuchsian surface subgroup. Recall that every hyperbolic manifold M3 can be represented as the quotient M3 = H3=G, where H3 is the hyperbolic 3-ball and G a Kleinian group. Using geometry and relying on fine statistical prop- erties of the frame flow on hyperbolic manifolds we proved in [14] the following result which implies the Surface Subgroup Theorem: Theorem 1.2 (Kahn-Markovic). Let M3 = H3=G denote a closed hy- perbolic 3-manifold. Given any > 0, there exists a (1+)-quasifuchsian group G < G. (Recall that a group is K-quasifuchsian if it is K-quasiconformal deformation of a Fuchsian group.) The nearly geodesic surfaces we constructed in [14] have large genus 1 (it can be shown that the genus of S grows polynomially with ). Moreover, each such surface f(S) ⊂ M3 represents the trivial homology 3 class in H2(M ; Z). JK is supported by NSF grant number DMS 0905812; VM is supported by NSF grant number DMS 1201463 2000 Mathematics Subject Classification. Primary 20H10. 1 2 KAHN AND MARKOVIC A three holed sphere with a hyperbolic metric and geodesic boundary is called a pair of pants (after Thurston). Given R > 0, the pair of pants whose all 3 cuffs have the same length 2R is called the R-perfect pair of pants. Let S(R) denote a genus 2 Riemann surface that is obtained by gluing two R-perfect pairs of pants along their cuffs with the twist of +1. The induced orbifold is denoted by Orb(R). Theorem 1.2 is proved by the showing that for a given closed hy- perbolic 3-manifold M3 and for every > 0 and every large enough R, there exists a Riemann surface S = S(, R) and a continuous map f : S ! M3 such that the induced map between universal covers @f : @H2 ! @H3 is (1 + )-quasisymmetric. Moreover, the surface S admits a decomposition into R-perfect pants that are glued to each other with the twist of +1. In particular, such a Riemann surface S is a regular holomorphic cover of the Model Orbifold Orb(R). For fixed , R > 0, a good pair of pants (in a given hyperbolic 3- manifold) is a pair of pants whose cuffs have complex half-length close to R (see Section 3). In order to find the map f : S ! M3 one is guided by the following principles: (1) Do not start by trying to specify the surface S. (2) Instead, consider the good pants Π immersed in M3 as the building blocks and eventually construct the surface f(S) ⊂ M3 by appropriately assembling together all the good pants from Π. Typically there are many ways in which one can assemble the pants and get an immersed surface. Consider any finite formal sum W 2 NΠ. Taking two copies of each pair of pants (with opposite orientations) one obtains the new formal sum 2W 2 NΠ. Then along every geodesic in M3 that appears as a boundary curve of some of the pants from 2W one can pair off the pairs of pants that contain that geodesic as a boundary component (there may be many ways in which one can pair off these pants and we choose one way of doing it for each such geodesic). One can now assemble these pairs of pants according to the instructions to construct a closed surface in M3. So, we have constructed a closed surface S and a map f : S ! M3, but the induced map between fundamental groups is not necessarily injective. For example, if W denotes a single pair of good pants then the surface S is a genus two surface obtained by gluing together two pairs of pants. However, the corresponding map f : S ! M3 collapses one pair of pants onto the other and therefore the induced map between fundamental groups is not injective. THE SURFACE SUBGROUP AND THE EHRENEPREIS CONJECTURES 3 Observe that every such surface f(S) ⊂ M3 represents the trivial 3 homology class in H2(M ; Z). This is because each pair of pants from W is used twice and with different orientations. It is clear from the previous discussion that if one wants to glue pairs of good pants in M3 to get a nearly geodesic surface (and thus a quasifuchsian surface) then any two pairs of pants that are glued along a geodesic should meet at an angle that is close to π. It turns out that in order to assemble good pants and construct a nearly geodesic surface in M3, what is needed is that the good pants are equidistributed in M3, which follows from the exponential mixing of the frame flow. We will explain this in more details in the next section, but here we state the mixing principle: Lemma 1.1 (Exponential Mixing). Let M3 denote a closed hyperbolic manifold (in particular a hyperbolic surface). There exists q > 0 that depends only on M3 such that the following holds. Let ; φ : F(M3) ! R be two C1 functions (here F(M3) denotes the frame bundle, if M3 is a Riemann surface this is just the tangent bundle). Then assuming that the volume of the frame bundle F(M3) is equal to 1, for every r 2 R the inequality Z Z Z ∗ −qjrj (gr )(x)φ(x) dΛ(x) − (x) dΛ(x) φ(x) dΛ(x) ≤ Ce ; F(M3) F(M3) F(M3) holds, where C > 0 only depends on the C1 norm of and φ. 2. The Ehrenpreis Conjecture The Ehrenpreis conjecture was an old conjecture in the theory of Riemann surfaces. The idea is that although two Riemann surfaces S and T do not have a common finite cover (all covers in this proposal are regular and unbranched) one should still be able to interpolate between certain finite covers of S and T respectively (according to Gromov this statement goes back to Riemann). The precise formulation of the conjecture is as follows: Conjecture 2.1 (Ehrenpreis Conjecture). Let S and T denote two closed Riemann surfaces of genus at least 2 and let > 0. Then there exists finite covers S1 and T1 of S and T respectively, such that S1 and T1 are (1 + ) quasiconformal to each other (that is there exists (1 + )-quasiconformal map f : S1 ! T1). In [16] J. Kahn and I have announced a proof of this conjecture. 4 KAHN AND MARKOVIC Remark. The Enhrenpreis Conjecture is harder to prove because there may be more pants on one side of a closed geodesic than the other. So we need to add in a signed sum of pants so that there are an equal number on both sides of every good geodesic. Computing this correction requires the \good pants homology", which we develop in [16]. In fact, we prove this conjecture by proving the statement that every closed hyperbolic Riemann surface has a virtual decomposition into good pairs of pants that are glued by a twist that is nearly equal to +1. Recall that (, R)-good pair of pants is a pair of pants whose cuffs have the length -close to R. We prove the following virtual decomposition type theorem: Theorem 2.1 (Kahn-Markovic). Let S be a hyperbolic surface and let > 0. Then for every large enough R > 0, the surface S has a finite cover S1 that can be decomposed into (, R)-good pants such that every two adjacent pairs of good pants are glued with the twist that is R close to +1. We then show that this surface S1 is quasiconformally close to a finite cover of the model orbifold Orb(R) (see above for the definition of the Model orbifold Orb(R)): Theorem 2.2 (Kahn-Markovic). Let S be a closed hyperbolic Riemann surface. Then for every K > 1, and every large enough R > 0 there are finite covers S1 and O1 of the surface S and the model orbifold Orb(R) respectively, and a K-quasiconformal map f : S1 ! O1. The Ehrenpreis conjecture is an immediate corollary of this theorem. Theorem 2.3 (Kahn-Markovic). Let S and M denote two closed Rie- mann surfaces. For any K > 1, one can find finite degree covers S1 and M1 of S and M respectively, such that there exists a K-quasiconformal map f : S1 ! M1. The proof of the above Virtual Decomposition theorem follows from the equidistribution of the good pants (in much the same way as in the proof of the Surface Subgroup Theorem) and from the Correction theory that we will outline below. 3. The Setup and Main Ideas 3.1. The feet of a pair of pants. A pair of pants is a compact hyper- bolic Riemann surface with geodesic boundary that is homeomorphic to the sphere minus three disjoint round open disks. Any such pair of pants is determined by the lengths of the three boundary components, THE SURFACE SUBGROUP AND THE EHRENEPREIS CONJECTURES 5 which are called cuffs.