arXiv subjects

Jeffrey Brock

Publications and source records attributed to Jeffrey Brock.

20 records · Page 2Linked to original sources

arXiv2003

Tameness on the boundary and Ahlfors' measure conjecture

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compression body, or (3) N is a strong limit of geometrically finite manifolds. The first case proves Ahlfors' measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit G of geometrically finite Kleinian groups, the limit set of G is either of Lebesgue measure zero or all of the Riemann sphere. Thus, Ahlfors' conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.

math.GT↗

arXiv2001

Curvature and rank of Teichmüller space

Let S be a surface with genus g and n boundary components and let d(S) = 3g-3+n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmuller space Teich(S) is Gromov-hyperbolic if and only if d(S) <= 2. When d(S) >= 3 the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of R^k, k >= 2); when d(S) <= 2 we combine the hyperbolicity of the complex of curves and the relative hyperbolicity of CP(S) prove Gromov-hyperbolicity. We prove moreover that Teich(S) admits no geodesically complete Gromov-hyperbolic metric of finite covolume when d(S) >= 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) >= 2.

math.GT↗