Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of hyperbolic 3-manifolds

This paper proves a theorem about Dehn surgery using a new theorem about PSL2C character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictly between r − 1 and r + 1. A corollary is that any small knot

Let N = H 3 =\Gamma be a hyperbolic 3-manifold with free fundamental group 1 (N) = \Gamma = hA; Bi; such that [A; B] is parabolic. We show that the limit set of N is always locally connected. More precisely,

We study the spectrum of the Dirac operator on hyperbolic manifolds of finite volume. Depending on the spin structure it is either discrete or the whole real line. For link complements in S 3 we give a simple criterion in terms of linking numbers for when essential spectrum can occur. We compute the accumulation rate of the eigenvalues of a sequence of closed hyperbolic 2- or 3-manifolds degenerating into a noncompact hyperbolic manifold of finite volume. It turns out that in three dimensions there is no clustering at all.

Given a closed hyperbolic n-manifold M, we study the flat Lorentzian structures on M × R such that M ×{0} is a Cauchy surface. We show there exist only two maximal structures sharing a fixed holonomy (one future complete and the other one past complete). We study the geometry of those maximal spacetimes in terms of cosmological time. In particular, we study the asymptotic behaviour of the level surfaces of the cosmological time. As a by-product, we get that no affine deformation of the hyperbolic holonomy ρ: π1(M) → SO(n, 1) of M acts freely and properly on the whole Minkowski space. The present work generalizes the case n = 2 treated by Mess, taking from a work of Benedetti and Guadagnini the emphasis on the fundamental rôle played by the cosmological time. In the last sections, we introduce measured geodesic stratifications on M, that in a sense furnish a good generalization of measured geodesic laminations in any dimension and we investigate relationships between measured stratifications on M and Lorentzian structures on M × R. 1.

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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We employ the deformation theory of hyperbolic cone-manifolds to prove that N is tame if it has non-empty conformal boundary. The theorem proves Ahlfors’ measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit Γ of geometrically finite Kleinian groups, the limit set of Γ is either of Lebesgue measure zero or all of Ĉ. Thus, Ahlfors ’ conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston. We then employ our techniques to show that each strong limit of geometrically finite hyperbolic 3-manifolds is tame and to verify a conjectural picture of the deformation space due to Thurston. 1

This paper investigates the behavior of the Hausdorff dimensions of the limit sets n and of a sequence of Kleinian groups \Gamma n ! \Gamma, where M = H 3 =\Gamma is geometrically finite. We show if \Gamma n ! \Gamma strongly, then: (a) Mn = H 3 =\Gamma n is geometrically finite for all n AE 0, (b) n ! in the Hausdorff topology, and (c) H: dim( n ) ! H: dim(), if H: dim() 1. On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when H: dim() ! 1. Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higher-dimensional manifolds. Applications are given to quasifuchsian groups and their limits.

This paper gives examples of hyperbolic 3-manifolds whose SL(2,C) character varieties have ideal points for which the associated roots of unity are not ±1. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to whether roots of unity other than ±1 occur. 1