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)



Entropy, Weil-Petersson translation distance and Gromov norm for surface automorphisms

Author: Sadayoshi Kojima
Journal: Proc. Amer. Math. Soc. 140 (2012), 3993-4002
MSC (2010): Primary 37E30; Secondary 57M27, 57M50
Published electronically: March 28, 2012
MathSciNet review: 2944738
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Thanks to a theorem of Brock on the comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of a Gromov norm of its mapping torus from below and an inbounded geometry case from above. We also prove that the Weil-Petersson translation distance does the same from both sides in general. The proofs are in fact immediately derived from the theorem of Brock, together with some other strong theorems and small observations.

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

  • 1. R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. MR 0175106 (30:5291)
  • 2. L. Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta. Math., 141 (1978), 73-98. MR 0477161 (57:16704)
  • 3. M.  Bridson and A. Haefliger, Metric spaces of non-positive curvature, Comprehensive Studies in Math., vol. 319, Springer (1999). MR 1744486 (2000k:53038)
  • 4. J. Brock, Weil-Petersson translation distance and volumes of mapping tori, Communications in Analysis and Geometry, 11 (2003), 987-999. MR 2032506 (2004k:32018)
  • 5. J. Brock, H. Masur and Y. Minsky, Asymptotics of Weil-Petersson geodesics II: Bounded geometry and unbounded entropy, arXiv:math.GT/1004.4401.
  • 6. G. Daskalopoulis and R. Wentworth, Classification of Weil-Petersson isometries, Amer. J. Math., 125 (2003), 941-975. MR 1993745 (2004d:32011)
  • 7. A. Fathi, F. Laudenbach and V. Poenaru, Travaux de Thurston sur les surfaces, Asterisque, 66-67, Société Mathématique de France, Paris (1979). MR 568308 (82m:57003)
  • 8. F. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, Volume 76, Amer. Math. Soc. (2000). MR 1730906 (2001d:32016)
  • 9. E. Kin, S. Kojima and M. Takasawa, Entropy versus volume for pseudo-Anosovs, Experimental Math., 18 (2009), 397-407. MR 2583541 (2010j:37064)
  • 10. M. Linch, A comparison of metrics on Teichmüller space, Proc. Amer. Math. Soc., 43 (1974), 349-352. MR 0338453 (49:3217)
  • 11. H. Masur, The existence of the Weil-Petersson metric to the boundary of Teichmüller space, Duke Math. J., 43 (1976), 623-635. MR 0417456 (54:5506)
  • 12. H. Masur and M. Wolf, The Weil-Petersson isometry group, Geometriae Dedicata, 93 (2002), 177-190. MR 1934697 (2003j:32017)
  • 13. Y. Minsky, Teichmüller geodesics and ends of hyperbolic $ 3$-manifolds, Topology, 32 (1993), 625-647. MR 1231968 (95g:57031)
  • 14. D. Mumford, A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc., 28 (1971), 289-294. MR 0276410 (43:2157)
  • 15. J.-P. Otal and L. Kay, The hyperbolization theorem for fibered $ 3$-manifolds, SMF/AMS Texts and Monographs, 7, American Mathematical Society (2001). MR 1855976 (2002g:57035)
  • 16. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
  • 17. G. Perelman, Ricci flow with surgery of three-manifolds, arXiv:math.DG/0303109.
  • 18. T. Soma, The Gromov invariant of links, Inventiones Math., 64 (1981), 445-454. MR 632984 (83a:57014)
  • 19. W. Thurston, The geometry and topology of $ 3$-manifolds, Lecture Notes, Princeton University (1979).
  • 20. W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc., 19 (1988), 417-431. MR 956596 (89k:57023)
  • 21. W. Thurston, Hyperbolic structures on $ 3$-manifolds II: Surface groups and $ 3$-manifolds which fiber over the circle, preprint.
  • 22. S. Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math., 61 (1975), 573-577. MR 0422692 (54:10678)
  • 23. S. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Inventiones Math., 85 (1986), 119-145. MR 842050 (87j:32070)
  • 24. S. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom., 25 (1987), 275-295. MR 880186 (88e:32032)
  • 25. S. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, Surveys in Differential Geometry, vol. VIII (2003), 357-393. MR 2039996 (2005h:32032)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37E30, 57M27, 57M50

Retrieve articles in all journals with MSC (2010): 37E30, 57M27, 57M50

Additional Information

Sadayoshi Kojima
Affiliation: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo 152-8552, Japan

Keywords: Surface automorphism, entropy, Teichmüller translation distance, Weil-Petersson translation distance, Gromov norm.
Received by editor(s): May 6, 2010
Received by editor(s) in revised form: March 6, 2011, and May 13, 2011
Published electronically: March 28, 2012
Additional Notes: The author was partially supported by Grant-in-Aid for Scientific Research (A) (No. 22244004), JSPS, Japan
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society