Geometry & Topology

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

Jeffrey Brock and Howard Masur

Full-text: Open access

Abstract

We analyze the coarse geometry of the Weil–Petersson metric on Teichmuller 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 nontrivial collection of subsets with respect to which it strongly relatively hyperbolic; this extends a theorem of Behrstock, Drutu and Mosher [submitted] in dimension 6 and higher into the intermediate range (it is hyperbolic if and only if the dimension is 1 or 2 by Brock and Farb [Amer. J. Math. 128 (2006) 1-22]). 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.

Article information

Source
Geom. Topol., Volume 12, Number 4 (2008), 2453-2495.

Dates
Received: 12 July 2007
Revised: 14 September 2008
Accepted: 10 July 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800130

Digital Object Identifier
doi:10.2140/gt.2008.12.2453

Mathematical Reviews number (MathSciNet)
MR2443970

Zentralblatt MATH identifier
1176.30096

Subjects
Primary: 30F60: Teichmüller theory [See also 32G15]
Secondary: 20F67: Hyperbolic groups and nonpositively curved groups

Keywords
Teichmüller space Weil–Petersson geometry pants graph relative hyperbolicity

Citation

Brock, Jeffrey; Masur, Howard. Coarse and synthetic Weil–Petersson geometry: quasi-flats, geodesics and relative hyperbolicity. Geom. Topol. 12 (2008), no. 4, 2453--2495. doi:10.2140/gt.2008.12.2453. https://projecteuclid.org/euclid.gt/1513800130


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References

  • J A Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006) 1523–1578
  • J A Behrstock, C Druţu, L Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, submitted
  • J A Behrstock, Y Minsky, Dimension and rank for mapping class groups, Ann. of Math. (2) 167 (2008) 1055–1077
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften [Fund. Princ. of Math. Sciences] 319, Springer, Berlin (1999)
  • J F Brock, The Weil–Petersson metric and volumes of $3$–dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16 (2003) 495–535
  • J F Brock, The Weil–Petersson visual sphere, Geom. Dedicata 115 (2005) 1–18
  • J F Brock, B Farb, Curvature and rank of Teichmüller space, Amer. J. Math. 128 (2006) 1–22
  • J F Brock, H A Masur, Y N Minsky, Asymptotics of Weil–Petersson geodesics I: ending laminations, recurrence, and flows
  • G Daskalopoulos, R Wentworth, Classification of Weil–Petersson isometries, Amer. J. Math. 125 (2003) 941–975
  • C Druţu, Relatively hyperbolic groups: geometry and quasi-isometric invariance
  • C Druţu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959–1058 With an appendix by Denis Osin and Sapir
  • A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997) 653–692
  • B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810–840
  • A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221–237
  • Y Imayoshi, M Taniguchi, An introduction to Teichmüller spaces, Springer, Tokyo (1992) Translated and revised from the Japanese by the authors
  • E Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint (1999)
  • B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997) 115–197
  • I Kra, On the Nielsen–Thurston–Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981) 231–270
  • H A Masur, Y N Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999) 103–149
  • H A Masur, Y N Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902–974
  • A Putman, A note on the connectivity of certain complexes associated to surfaces, preprint
  • K Rafi, A combinatorial model for the Teichmüller metric, Geom. Funct. Anal. 17 (2007) 936–959
  • S Schleimer, Notes on the complex of curves, unpublished
  • S A Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987) 275–296
  • S A Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space, from: “Surveys in differential geometry, Vol. VIII (Boston, MA, 2002)”, Surv. Differ. Geom. VIII, Int. Press, Somerville, MA (2003) 357–393