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

By Jeffrey F. Brock. 2001

206 citations recorded by the source corpus.

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

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6fb8d42777f69296e98551fba8d9708b5132f811

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

3‐manifolds efficiently bound 4‐manifolds (2005, background): “In addition, we give an O(n4 ) bound for the complexity of a spin 4-manifold bounding a given spin 3-manifold.”
Asymptotic Geometry of the Mapping Class Group and Teichmuller Space (2005, methodology): “21 [14] P(S) is quasi-isometric to the Teichmüller space of S with the Weil–Petersson metric.”
Quasiflats in hierarchically hyperbolic spaces (2017, methodology): “rem A is to the Weil-Petersson metric on Teichmuller space, which was proven to be an asymphoric HHS by virtue of Brock’s theorem that the pants graph is quasi-isometric to the Weil-Petersson metric [Bro03] and results of [Beh06, BKMM12, MM00], for details see [BHS17b, Theorem G]. Brock asked whethe”
Families of Riemann Surfaces and Weil-Petersson Geometry (2010, background): “ing a single curve with a once or twice intersecting curve (a Hatcher-Thurston move). Brock discovered that Teichmu¨ller space with the WP metric is quasi isometric to P(R) with the unit-edge metric, [Bro03]. At large scale on Teichmu¨ller space WP distance is combinatorially determined. The importa”
Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows (2008, methodology): “A quasi-isometric model was obtained for the Weil-Petersson metric in [ Br1 ] using pants decompositions of surfaces.”
Generalized Markoff maps and McShane's identity (2005, background): “G T ] 2 2 Fe b for closed hyperbolic 3-manifolds which are obtained by doing hyperbolic Dehn surgery on such manifolds.”
Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity (2007, methodology): “S/, which carries a convenient coarse description in terms of combinatorics of pants decompositions of a surface [5].”
Geometry of the Weil-Petersson completion of Teichm\ (2005, background): “Brock considered the metric space, the pants graph CP(F), having vertices the distinct pants decompositions of F and joining edges of unit-length for pants decompositions differing by a single elementary move,[ 8 ].”
Behavior of geodesic-length functions on Teichmüller space (2007, background): “The above comparability on Bers regions provides for local considerations for Brock’s quasi isometry of the pants complex CP to Teichmüller space with the WP metic, [Bro03].”
A Combinatorial Model for the Teichmüller Metric (2005, methodology): “ space to combinatorial spaces. In [MM99] Masur and Minsky show that the electriﬁed Teichmu¨ller space is quasi-isometric to the complex of curves and therefore is also δ–hyperbolic. Brock has shown ([Bro03]) that Teichmu¨ller space equipped with the Weil-Petersson metric is quasi-isometric to the p”
Automorphisms of the pants complex (2004, background): “Brock proved that C(1)P(S) models the Teichmüller space endowed with the WeilPetersson metric, TW P(S), in that the spaces are quasi-isometric (see [1]).”