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)



Word length in surface groups with characteristic generating sets

Author: Danny Calegari
Journal: Proc. Amer. Math. Soc. 136 (2008), 2631-2637
MSC (2000): Primary 57M07
Published electronically: February 29, 2008
MathSciNet review: 2390536
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length of any element with a nonzero crossing number is positive, and bounded below by a constant depending only (and explicitly) on a bound on the crossing numbers of generating elements. This answers a question of Benson Farb.

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

  • 1. Christophe Bavard, Longueur stable des commutateurs, Enseign. Math. (2) 37 (1991), no. 1-2, 109–150 (French). MR 1115747
  • 2. Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
  • 3. Robert Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 53–63. MR 624804
  • 4. D. Calegari, scl, monograph; draft available for download from the author's webpage˜dannyc/scl/toc.html.
  • 5. D. Calegari and K. Fujiwara, Stable commutator length in word-hyperbolic groups, preprint, math.GR/0611889.
  • 6. David B. A. Epstein and Koji Fujiwara, The second bounded cohomology of word-hyperbolic groups, Topology 36 (1997), no. 6, 1275–1289. MR 1452851, 10.1016/S0040-9383(96)00046-8
  • 7. M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, 10.1007/978-1-4613-9586-7_3

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M07

Retrieve articles in all journals with MSC (2000): 57M07

Additional Information

Danny Calegari
Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125

Received by editor(s): May 21, 2007
Published electronically: February 29, 2008
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.