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Stable commutator length is rational in free groups

Author: Danny Calegari
Journal: J. Amer. Math. Soc. 22 (2009), 941-961
MSC (2000): Primary 57M07, 20F65, 20J05
Published electronically: May 1, 2009
MathSciNet review: 2525776
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Abstract: For any group, there is a natural (pseudo-)norm on the vector space $ B_1^H$ of real homogenized (group) $ 1$-boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of this pseudo-norm is a rational polyhedron.

It follows that the stable commutator length in free groups takes on only rational values. Moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group.

The proof of these facts yields an algorithm to compute the stable commutator length in free groups. Using this algorithm, we answer a well-known question of Bavard in the negative, constructing explicit examples of elements in free groups whose stable commutator length is not a half-integer.

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. Abdessalam Bouarich, Suites exactes en cohomologie bornée réelle des groupes discrets, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 11, 1355–1359 (French, with English and French summaries). MR 1338286
  • 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. Danny Calegari, Foliations and the geometry of 3-manifolds, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2007. MR 2327361
  • 5. Danny Calegari, Surface subgroups from homology, Geom. Topol. 12 (2008), no. 4, 1995–2007. MR 2431013, 10.2140/gt.2008.12.1995
  • 6. D. Calegari, Faces of the scl norm ball, Geom. Top. 13 (2009), 1313-1336
  • 7. D. Calegari, Scl, sails and surgery, preprint, in preparation
  • 8. D. Calegari, scallop, computer program, available from $ \sim$dannyc
  • 9. D. Calegari, scl, monograph, to appear in Mathematical Society of Japan Monographs; available from$ \sim$dannyc
  • 10. D. Calegari and K. Fujiwara, Stable commutator length in word hyperbolic groups, Groups, Geometry, Dynamics, to appear.
  • 11. George B. Dantzig, Linear programming and extensions, Princeton University Press, Princeton, N.J., 1963. MR 0201189
  • 12. David Gabai, Foliations and the topology of 3-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
  • 13. C. Gordon and H. Wilton, On surface subgroups of doubles of free groups, preprint, arXiv:0902.3693
  • 14. Michael Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99 (1983). MR 686042
  • 15. M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
  • 16. John Hempel, 3-Manifolds, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1976. Ann. of Math. Studies, No. 86. MR 0415619
  • 17. M. Kiyomi, exlp, computer program, available at masashi777/exlp.html
  • 18. A. Makhorin, glpsol, computer program, available from
  • 19. Lee Mosher and Ulrich Oertel, Two-dimensional measured laminations of positive Euler characteristic, Q. J. Math. 52 (2001), no. 2, 195–216. MR 1838363, 10.1093/qjmath/52.2.195
  • 20. R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770
  • 21. Ulrich Oertel, Homology branched surfaces: Thurston’s norm on 𝐻₂(𝑀³), Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 253–272. MR 903869
  • 22. Richard Rannard, Computing immersed normal surfaces in the figure-eight knot complement, Experiment. Math. 8 (1999), no. 1, 73–84. MR 1685039
  • 23. Peter Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555–565. MR 0494062
  • 24. Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
  • 25. William P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i–vi and 99–130. MR 823443
  • 26. Dongping Zhuang, Irrational stable commutator length in finitely presented groups, J. Mod. Dyn. 2 (2008), no. 3, 499–507. MR 2417483, 10.3934/jmd.2008.2.499

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Additional Information

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

Received by editor(s): February 18, 2008
Published electronically: May 1, 2009
Dedicated: Dedicated to Shigenori Matsumoto on the occasion of his 60th birthday
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.