Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Constructing convex planes in the pants complex


Authors: Javier Aramayona, Hugo Parlier and Kenneth J. Shackleton
Journal: Proc. Amer. Math. Soc. 137 (2009), 3523-3531
MSC (2000): Primary 57M50; Secondary 05C12
DOI: https://doi.org/10.1090/S0002-9939-09-09907-9
Published electronically: June 29, 2009
Previous version: Original version posted May 15, 2009
Corrected version: Current version corrects publisher's omission of labels in Figure 2
MathSciNet review: 2515421
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Our main theorem identifies a class of totally geodesic subgraphs of the $ 1$-skeleton of the pants complex, referred to as the pants graph, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the pants graph of any surface of complexity at least $ 3$.


References [Enhancements On Off] (What's this?)

  • 1. J. Aramayona, H. Parlier, K. J. Shackleton, Totally geodesic subgraphs of the pants complex, Math. Res. Lett. 15, no. 2-3 (2008), 309-320. MR 2385643 (2008m:57004)
  • 2. J. A. Behrstock, C. Druţu, L. Mosher, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, to appear in Math. Annalen.
  • 3. J. F. Brock, The Weil-Petersson metric and volumes of $ 3$-dimensional hyperbolic convex cores, J. Amer. Math. Soc. 16, no. 3 (2003), 495-535. MR 1969203 (2004c:32027)
  • 4. J. F. Brock, B. Farb, Curvature and rank of Teichmüller space, Amer. J. Math., 128, no. 1 (2006), 1-22. MR 2197066 (2006j:32013)
  • 5. J. F. Brock, H. A. Masur, Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity, Geom. Topol. 12 (2008), no. 4, 2453-2495. MR 2443970
  • 6. T. Chu, The Weil-Petersson metric in the moduli space, Chinese J. Math. 4 (1976), no. 2, 29-51. MR 0590105 (58:28683)
  • 7. A. E. Hatcher, W. P. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), 221-237. MR 0579573 (81k:57008)
  • 8. D. Margalit, Automorphisms of the pants complex, Duke Math. J. 121, no. 3 (2004), 457-479. MR 2040283 (2004m:57037)
  • 9. H. A. Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43, no. 3 (1976), 623-635. MR 0417456 (54:5506)
  • 10. H. A. Masur, Y. N. Minsky, Geometry of the complex of curves II: Hierarchical structure, Geom. Funct. Anal. 10, no. 4 (2000), 902-974. MR 1791145 (2001k:57020)
  • 11. H. A. Masur, S. Schleimer, The pants complex has only one end, ``Spaces of Kleinian groups'' (eds. Y. N. Minsky, M. Sakuma, C. M. Series), Math. Soc. Lecture Note Ser. 329 (2006), 209-218. MR 2258750 (2007g:57035)
  • 12. S. A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275-296. MR 880186 (88e:32032)
  • 13. S. A. Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975), no. 2, 573-577. MR 0422692 (54:10678)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M50, 05C12

Retrieve articles in all journals with MSC (2000): 57M50, 05C12


Additional Information

Javier Aramayona
Affiliation: Department of Mathematics, National University of Ireland, Galway, Ireland
Email: Javier.Aramayona@nuigalway.ie

Hugo Parlier
Affiliation: Institute of Geometry, Algebra and Topology, École Polytechnique Fédérale de Lausanne, Bâtiment BCH, CH-1015 Lausanne, Switzerland
Email: hugo.parlier@a3.epfl.ch

Kenneth J. Shackleton
Affiliation: University of Tokyo IPMU, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 277-8568, Japan
Email: kenneth.shackleton@ipmu.jp, kjs2006@alumni.soton.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-09-09907-9
Keywords: Pants complex, Weil-Petersson metric
Received by editor(s): February 27, 2007
Received by editor(s) in revised form: October 26, 2008
Published electronically: June 29, 2009
Additional Notes: The first author was partially supported by a short-term research fellowship at the Université de Provence, and the third author by a long-term JSPS postdoctoral fellowship, number P06034
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society