Yair N. Minsky

GEOMETRY OF THE COMPLEX OF CURVES II: HIERARCHICAL STRUCTURE

Abstract. ((Without Abstract)).

The classification of punctured-torus groups

Thurston’s ending lamination conjecture proposes that a flnitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for punctured-torus groups. These are free two-generator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of punctured-torus groups (including Bers’ conjecture that the quasi-Fuchsian groups are dense in this space) and prove a rigidity theorem: two punctured-torus groups are quasi-conformally conjugate if and only if they are topologically conjugate.

Rank-1 phenomena for mapping class groups

Let Σg be a closed, orientable, connected surface of genus g ≥ 1. The mapping class group Mod(Σg) is the group Homeo(Σg)/Homeo0(Σg) of isotopy classes of orientation-preserving homeomorphisms of Σg. It has been a recurring theme to compare the group Mod(Σg) and its action on the Teichmuller space T (Σg) to lattices in simple Lie groups and their actions on the associated symmetric spaces. Indeed, the groups Mod(Σg) share many of the properties of (arithmetic) lattices in semisimple Lie groups. For example they satisfy the Tits alternative, they have finite virtual cohomological dimension, they are residually finite, and each of their solvable subgroups is polycyclic. A well-known dichotomy among the lattices in simple Lie groups is between lattices in rank one groups and higher-rank lattices, i.e. those lattices in simple Lie groups of R-rank at least two. It is somewhat mysterious whether Mod(Σg) is similar to the former or the latter. Some higher rank behavior of Mod(Σg) is indicated by the cusp structure of moduli space, by the fact that Mod(Σg) has Serre’s property (FA) [CV], and by Ivanov’s version (see, e.g. [Iv2]) for Mod(Σg) of Tits’s Theorem on automorphism groups of higher rank buildings. In this note we add two more properties to the list (see, e.g. [Iv1, Iv2, Iv3] and the references therein) of properties which exhibits similarities of Mod(Σg) with lattices in rank one groups: every infinite order element of Mod(Σg) has linear growth in the word metric, and Mod(Σg) is not bound∗Supported in part by NSF grant DMS 9704640 and by a Sloan Foundation fellowship. †Supported in part by the US-Israel BSF grant. ‡Supported in part by NSF grant DMS 9971596

Teichmüller geodesics and ends of hyperbolic 3-manifolds

THE THEORY of the structure and deformations of hyperbolic 3-manifolds depends in an essential way on a good understanding of the geometry of incompressible maps of surfaces into these manifolds. Consider, for example, a hyperbolic 3-manifold N homeomorphic to S x R where S is a closed surface of genus g > 1. Thurston and Bonahon showed how to fill the convex hull of N with “pleated surfaces” homotopic to the obvious map S -+ S x {0} (see Sections 2.3, 2.4) and these surfaces have induced metrics which determine points in the Teichmiiller space r(S) of conformal (or hyperbolic) structures on S. It has been conjectured that the locus of these points is related in an approximate way to a geodesic in F(S), and this is known to be true for a class of examples arising from hyperbolic structures on surface bundles over a circle (see [S]). This kind of information has implications concerning the geometry of N, which will be discussed more fully in a forthcoming paper ([25]). From a more differential-geometric point of view, one can consider, for any metric 0 on S, a mapf, : S -+ N of least “energy” (see Section 3) in the above homotopy class. This least energy is then some function 8(c), andf, is a harmonic map. One can then ask about the locus of points [o] in 3(S) where 8 is bounded above by a given constant, and its relationship to the above set of metrics induced from pleated surfaces. We give here a proof of the following result in this direction, where the crucial restrictive hypothesis we need to make is a positive lower bound on the injectivity radius inj,(x) for all x E N.

Bounded geometry for Kleinian groups

Abstract.We show that a Kleinian surface group, or hyperbolic 3-manifold with a cusp-preserving homotopy-equivalence to a surface, has bounded geometry if and only if there is an upper bound on an associated collection of coefficients that depend only on its end invariants. Bounded geometry is a positive lower bound on the lengths of closed geodesics. When the surface is a once-punctured torus, the coefficients coincide with the continued fraction coefficients associated to the ending laminations.

High distance knots

We construct knots in S-3 with Heegaard splittings of arbitrarily high distance, in any genus. As an application, for any positive integers t and b we find a tunnel number t knot in the three-sphere which has no (t, b) -decomposition.

Kleinian groups and the complex of curves

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for the Kleinian group to have ‘bounded geometry’ (lower bounds on injectivity radius) in terms of a sequence of coecients (subsurface projections) computed using the ending invariants of the group and the complex of curves. These results are directly analogous to those obtained in the case of puncturedtorus surface groups. In that setting the ending invariants are points in the closed unit disk and the coecients are closely related to classical continuedfraction coecients. The estimates obtained play an essential role in the solution of Thurston’s ending lamination conjecture in that case.

Centroids and the rapid decay property in mapping class groups

We study a notion of an equivariant, Lipschitz, permutation- invariant centroid for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).

Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds

Let M be a compact 3-manifold whose interior admits a complete hyperbolic structure. We let Λ(M) be the supremum of λ0(N) where N varies over all hyperbolic 3-manifolds homeomorphic to the interior of N. Similarly, we let D(M) be the infimum of the Hausdorff dimensions of limit sets of Kleinian groups whose quotients are homeomorphic to the interior of M. We observe that A(M) = D(M)(2 − D(M)) if M is not handlebody or a thickened torus. We characterize exactly when A(M) = 1 and D(M) = 1 in terms of the characteristic submanifold of the incompressible core of M.

Harmonic maps into hyperbolic 3-manifolds

High-energy degeneration of harmonic maps of Riemann surfaces into a hyperbolic 3-manifold target is studied, generalizing results of [M 1] in which the target is two-dimensional. The Hopf foliation of a high-energy map is mapped to an approximation of its geodesic representative in the target, and the ratio of the squared length of that representative to the extremal length of the foliation in the domain gives an estimate for the energy. The images of harmonic maps obtained when the domain degenerates along a Teichmuller ray are shown to converge generically to pleated surfaces in the geometric topology or to leave every compact set of the target when the limiting foliation is unrealizable