Jeffrey F. Brock

The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

We present a coarse interpretation of the Weil-Petersson distance dWP(X,Y ) between two finite area hyperbolic Riemann surfaces X and Y using a graph of pants decompositions introduced by Hatcher and Thurston. The combinatorics of the pants graph reveal a connection between Riemann surfaces and hyperbolic 3-manifolds conjectured by Thurston: the volume of the convex core of the quasi-Fuchsian manifold Q(X,Y ) with X and Y in its conformal boundary is comparable to the Weil-Petersson distance dWP(X,Y ). In applications, we relate the Weil-Petersson distance to the Hausdorff dimen- sion of the limit set and the lowest eigenvalue of the Laplacian for Q(X,Y ), and give a new finiteness criterion for geometric limits. Mathematics Department, University of Chicago, 5734 S. University Ave., Chicago, IL 60637 E-mail address: brock@math.uchicago.edu

Almost Alternating Links

Abstract We introduce the category of almost alternating links: nonalternating links which have a projection for which one crossing change yields an alternating projection. We extend this category to m -almost alternating links which require m crossing changes to yield an alternating projection. We show that all but five of the nonalternating knots up through eleven crossings and links up through ten crossing are almost alternating. We also prove that a prime almost alternating knot is either a hyperbolic knot or a torus knot. We then obtain a bound on the span of the bracket polynomial for m -almost alternating links and discuss applications.

On the density of geometrically finite Kleinian groups

The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3-manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends.

Continuity of Thurston's length function

Abstract. A measured lamination μ geodesically realized in a hyperbolic 3-manifold M has a well-defined average length, due to W. Thurston. For

Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity

M \cong S \times {\Bbb R}

WEIL-PETERSSON ISOMETRIES VIA THE PANTS COMPLEX

we prove that the function measuring the average length of the maximal realizable sublamination of μ varies bicontinuously in M and μ. Since connected, positive, non-realizable measured laminations arise as zeros of the length function, its continuity suggests new behavioral features of quasi-isometry invariants under limits of hyperbolic 3-manifolds.

Iteration of mapping classes and limits of hyperbolic 3-manifolds

We analyze the coarse geometry of the Weil-Petersson metric on Teichm¨ uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of the Weil-Petersson metric via consideration of its coarse quasi-isometric model, the pants graph. We show that in dimension 3 the pants graph is strongly relatively hyperbolic with respect to naturally defined product regions and show any quasi-flat lies a bounded distance from a single product. For all higher dimensions there is no non-trivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of [BDM] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative stability of quasi-geodesics in dimensions up through 3 provide for a strong understanding of the behavior of geodesics and a complete description of the CAT 0 -boundary of the Weil-Petersson metric via curve-hierarchies and their associated boundary laminations.

Injectivity radii of hyperbolic integer homology 3-spheres

We extend a theorem of Masur and Wolf which states that given a finite area hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four times-punctured sphere, the once-punctured torus, and the twice-punctured torus.

Local topology in deformation spaces of hyperbolic 3-manifolds

Abstract.Let ϕ∈Mod(S) be an element of the mapping class group of a surface S. We classify algebraic and geometric limits of sequences {Q(ϕiX,Y)}i=1∞ of quasi-Fuchsian hyperbolic 3-manifolds ranging in a Bers slice. When ϕ has infinite order with finite-order restrictions, there is an essential subsurface Dϕ⊂S so that the geometric limits have homeomorphism type S×ℝ-Dϕ×{0}. Typically, ϕ has pseudo-Anosov restrictions, and Dϕ has components with negative Euler characteristic; these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit. We show there is an s≥1 depending on ϕ and bounded in terms of S so that {Q(ϕsiX,Y)}i=1∞ converges algebraically and geometrically, and we give explicit quasi-isometric models for the limits.

Boundaries of Teichmüller spaces and end-invariants for hyperbolic 3-manifolds

We construct hyperbolic integer homology 3-spheres where the injectivity radius is arbitrarily large for nearly all points of the manifold. As a consequence, there exists a sequence of closed hyperbolic 3-manifolds which Benjamini-Schramm converge to H^3 whose normalized Ray-Singer analytic torsions do not converge to the L^2-analytic torsion of H^3. This contrasts with the work of Abert et. al. who showed that Benjamini-Schramm convergence forces convergence of normalized betti numbers. Our results shed light on a conjecture of Bergeron and Venkatesh on the growth of torsion in the homology of arithmetic hyperbolic 3-manifolds, and we give experimental results which support this and related conjectures.