HYPERBOLIC GEOMETRY AND HEEGAARD SPLITTINGS TALK GIVEN BY HOSSEIN NAMAZI 1. Preliminary Remarks The lecture is based on joint work with J. Brock, Y. Minsky and J. Souto. Example. Suppose H + and H are homeomorphic handlebodies of genus g 2, and we let f be a homeomorphism between the boundaries. We write M n = H + S H, with S = H + = H. We assume f is a pseudo-anosov homeomorphism and is generic. Here generic is chosen to mean that the stable and unstable laminations of f are not limits of meridians of H + and H. (A meridian is an essential simple closed curve on the boundary of a handlebody that is homotopically trivial in the handlebody.) Topological properties of the splitting imply that for n large, M n is atoroidal and is not Seifert fibered. Hence assuming the Geometrization Conjecture, one implies that M n admits a hyperbolic metric. But how does the metric look like? Question (Effective Geometrization). How a topological description of a 3-manifold describes the geometric picture? In the case of the examples (M n ), we have: Theorem 1 (Namazi-Souto). Given f and ɛ > 0, there exists n f,ɛ such that for n n f,ɛ, we have M n admits a Riemannian metric with curvature pinched in [ 1 ɛ, 1 + ɛ]. Even more, we obtain a decomposition of M n, equipped with this metric, which consist of a piece homeomorphic to S R and two end pieces homeomorphic to H and H +. The end pieces are bounded in a way that is independent of n and the metric on S R is (uniformly) bi-lipschitz to a large subset of the infinite cyclic cover of the mapping torus of f, equipped with the hyperbolic metric. Remark. It is a consequence of a theorem of Tian that in the above examples, the negatively curve metric is C 2 -close to (the) hyperbolic metric. Tian s theorem applies to all other negatively curved metrics that we obtain in our constructions. 2. Methods Suppose Y is a compact surface (possibly with boundary). An important object of study here is the complex of curves (and arcs) of Y denoted by C(Y ). We deal Date: Nov 12, 2007. 1
2 TALK GIVEN BY HOSSEIN NAMAZI only with the 1-skeleton of the complex which is defined as follows. The vertices are homotopy classes of non-peripheral simple closed curves and homotopy classes (rel ) of properly embedded essential arcs. Two vertices are joined by an edge if they can be realized disjointly. The complex is equipped with the graph metric. For a Heegaard splitting H + S H, the handlebodies H + and H naturally give rise to subcomplexes + and of C(S) which consist of meridians of H + and H respectively. Definition 1 (Hempel). The Heegaard distance HD associated to a Heegaard splitting is given by d C ( +, ). The above definition was motivated by the work of Casson-Gordon and there has been many topological investigation of this invariant. In particular, after putting together works of Haken, Casson-Grodon, Thompson, Hempel, Moriah-Schultens one has the following theorem: Theorem 2. If a closed 3-manifold M admits a splitting with HD 3 then M is irreducible, atoroidal and is not Seifert fibered. Motivated by these results Hempel conjectured: Conjecture (Hempel). Given g, if M admits a genus g splitting with HD sufficiently large then M is hyperbolic. The above conjecture must be correct because of the Geometrization Conjecture but that gives little insight into how the geometric structure looks like. As a consequence of our work, we give a proof of the above conjecture which does not use Perelman s proof of the Geometrization. In fact the proof is part of a program whose aim is to find bi-lipschitz models of the geometry of the 3-manifold depending only on the genus of the splitting. Let Y S be an essential non-annular subsurface. Let α, β C 0 (S) be in minimal position with respect to Y. We define d Y (α, β) = d C(Y ) (α Y, β Y ) when the above intersections are both nonempty and otherwise we do not define it. This definition can be extended to the case when Y is an annulus. In that case d Y (α, β) measures the difference between the twisting of α and β about the core of Y. For a precise definition and discussion see works of Masur-Minsky. Note that the definition easily extends to a case when α and β are collection of vertices of C(S). In that case d Y (α, β) = inf d Y (a, b). a α, b β 3. Bounded Geometry Definition 2. Given R 0, we say α and β have R-bounded combinatorics if d Y (α, β) R for all proper subsurfaces Y S. We say a Heegaard splitting
HYPERBOLIC GEOMETRY AND HEEGAARD SPLITTINGS 3 H + S H has R-bounded combinatorics if there exists pants decompositions P + + and P which have R-bounded combinatorics and d C (P +, P ) HD +R. Theorem 3 (Namazi). Given ɛ, R, g there exists d ɛ,r,g such that if M admits a genus g splitting with R-bounded combinatorics and HD d ɛ,r,g then M can be equipped with a Riemannian metric with curvature pinched in [ 1 ɛ, 1 + ɛ]. As a matter of fact, there is a decomposition of the manifold equipped with the described metric which is similar to the one we described for the first examples. The only difference here is that the metric on S R is different and is bi-lipschitz to the metric on a large subset of a hyperbolic manifold homeomorphic to S R with end invariants that can be described by using the combinatorics of the splitting. This certainly generalizes the previous theorem but still is not general because (1) The hyperbolic 3-manifolds obtained in this theorem all have ɛ-bounded geometry, i.e. a lower bound ɛ for the injectivity radius, where ɛ depends on R and g. (2) Even worse, these are not all the hyperbolic 3-manifolds with ɛ-bounded geometry and do not describe all possible pictures of such 3-manifolds. (3) Finally, an assumption of HD 1 is necessary in the hypothesis. However with Brock, Minsky and Souto, we could find a combinatorial condition that classifies all the hyperbolic 3-manifolds with a genus g splitting and a uniform lower bound for the injectivity radius. This is done by extending the bounded combinatorics condition for Heegaard splittings to another condition called generalized R-bounded combinatorics. Theorem 4 (Brock, Minsky, Namazi, Souto). Given ɛ and g there exists R such that a genus g splitting of a hyperbolic 3-manifold with ɛ-bounded geometry has to have generalized R-bounded combinatorics. Conversely given R and g there exists ɛ such that a hyperbolic 3-manifold with a genus g splitting with generalized R-bounded combinatorics has ɛ-bounded geometry. Even more the generalized R-bounded combinatorics provides a combinatorial diagram and from the diagram, we can construct a bi-lipschitz model for the hyperbolic metric with the bi-lispchitz constant depending only on R and g. 4. Hyperbolicity The aim of the program is to obtain models even when there is no lower bound for the injectivity radius. The work in that direction has been very promising and in particular, in answering the conjecture above we have: Theorem 5 (Brock, Minsky, Namazi, Souto). Given g, ɛ > 0, there exists d g,ɛ such that if M has a genus g splitting with HD d g,ɛ then M admits a Riemannian metric with curvature pinched in [ 1 ɛ, 1 + ɛ].
4 TALK GIVEN BY HOSSEIN NAMAZI We will sketch a proof of the above theorem. Suppose R > 0 and suppose M = H + S H has sufficiently large Heegaard distance. There are two cases to consider: Case 1. d Y ( +, ) R for every proper subsurface Y S with d C ( Y, + ) 3. Case 2. There exists Y S with d C ( Y, + ) 3 such that d Y ( +, ) > R. Briefly the idea is to show that in the fist case there is enough room to glue two handlebodies and produce negatively curved metrics. In the second case, we show there is a cusped hyperbolic 3-manifold with a hyperbolic Dehn surgery that gives back M. What allows us to use the number 3 for the cases described above and in the proof is a recent work of Masur-Schleimer. Among other things, they show that for a handlebody H with the set of meridians, and a subsurface Y H with d C ( Y, ) 3, we have diam Y ( ) C 0 ; also C 0 depends only on the genus of the handlebody. (We should remark that the proof does not really depend on this result of Masur-Schleimer but it gives more insight to what happens in more generality.) More precisely, in the first case, work of Masur-Minsky implies there exist pants decompositions µ +, µ with R-bounded combinatorics, such that d C (µ ±, ± ) 3 and µ +, µ are along a geodesic of minimal length that connects + and. Now we choose points τ +, τ in the Teichmüller space of S such that l τ ±(µ ± ) B 0 where B 0 is a constant depending only on g. For a handlebody of genus 2 very point τ of the Teichmüller space of H provides a geometrically finite structure on H with τ the conformal structure at infinity. Using this we take N +, N hyperbolic, geometrically finite structures on H +, H such that τ and τ + are the conformal structures at infinity respectively. Then we have the following claim: Claim 6. Given ɛ, g if HD 1 and N +, N are obtained as above then there exist compact subsets K ± N ± homeomorphic to S [0, 1] such that the composition S = S {0} S [0, 1] homeo. K ± N ± is in the isotopy class determined by S = H ±. Also there exists a diffeomorphism T : K + K that is ɛ-close (in C 3 -topology) to being an isometry and S [0, 1] homeo. K + T K is homotopic to the homeomorphism S [0, 1] K. Once the above claim is proved, we cut off the unbounded components of N + \ K + and N K and we glue the rest along by using the identification T : K + K. The obtained manifold is homeomorphic to M and a convex combination of the metrics on N + and N provides a metric that is ɛ-close to being hyperbolic. In the second case, the main idea is using the following: Theorem 7 (Hodgson-Kreckhoff). Suppose N is a hyperbolic 3-manifold with a rank 2 cusp and the torus T is the boundary of horospherical neighborhood of the cusp. Also let γ be an essential simple closed curve on T with normalized length at least 7.515, then N(γ) is hyperbolic, where N(γ) is the 3-manifold obtained by Dehn surgery on N along γ.
HYPERBOLIC GEOMETRY AND HEEGAARD SPLITTINGS 5 We have a subsurface Y S with d C ( Y, + ) 3 and d Y ( +, ) > R. We remove a regular neighborhood of Y from M and because of the condition d C ( Y, + ) the obtained 3-manifold M Y is atoroidal and has infinite π 1. So by Thurston s Hyperbolization theorem, its interior admits a finite volume hyperbolic metric. There is a canonical simple closed curve γ on the toroidal boundary of M \ Y such that M Y (γ) = M. We use d Y ( +, ) > R to show that for R large the normalized length of γ is at least 7.515 and therefore M = M Y (γ) is hyperbolic.