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Stable commutator length is rational in free groups

Author(s): Danny Calegari
Journal: J. Amer. Math. Soc. 22 (2009), 941-961.
MSC (2000): Primary 57M07, 20F65, 20J05
Posted: May 1, 2009
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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:

1.
C. Bavard, Longueur stable des commutateurs, Enseign. Math. (2), 37, 1-2, (1991), 109-150 MR 1115747 (92g:20051)

2.
A. Bouarich, Suites exactes en cohomologie bornée réelle des groupes discrets, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 11, 1355-1359 MR 1338286 (96c:55007)

3.
R. Brooks, Some remarks on bounded cohomology, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (SUNY Stony Brook NY 1978) Ann. Math. Stud. 97, Princeton Univ. Press 1981, 53-63 MR 624804 (83a:57038)

4.
D. Calegari, Foliations and the geometry of $ 3$-manifolds, Oxford Mathematical Monographs, Oxford Univ. Press, Oxford, 2007 MR 2327361 (2008k:57048)

5.
D. Calegari, Surface subgroups from homology, Geom. Top. 12 (2008), 1995-2007 MR 2431013 (2009d:20104)

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 http://www.its.caltech.edu/ $ \sim$dannyc

9.
D. Calegari, scl, monograph, to appear in Mathematical Society of Japan Monographs; available from http://www.its.caltech.edu/$ \sim$dannyc

10.
D. Calegari and K. Fujiwara, Stable commutator length in word hyperbolic groups, Groups, Geometry, Dynamics, to appear.

11.
G. Dantzig, Linear Programming and Extensions, Princeton Univ. Press, Princeton, 1963 MR 0201189 (34:1073)

12.
D. Gabai, Foliations and the topology of $ 3$-manifolds, J. Diff. Geom. 18 (1983), no. 3, 445-503 MR 723813 (86a:57009)

13.
C. Gordon and H. Wilton, On surface subgroups of doubles of free groups, preprint, arXiv:0902.3693

14.
M. Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. No. 56 (1982), 5-99 MR 686042 (84h:53053)

15.
M. Gromov, Asymptotic invariants of infinite groups, London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press, Cambridge, 1993 MR 1253544 (95m:20041)

16.
J. Hempel, $ 3$-Manifolds, Ann. Math. Stud. 86, Princeton Univ. Press, Princeton, 1976 MR 0415619 (54:3702)

17.
M. Kiyomi, exlp, computer program, available at http://members.jcom.home.ne.jp/ masashi777/exlp.html

18.
A. Makhorin, glpsol, computer program, available from http://www.gnu.org

19.
L. Mosher and U. Oertel, Two-dimensional measured laminations of positive Euler characteristic, Quart. J. Math. 52 (2001), 195-216 MR 1838363 (2002f:57061)

20.
R. Penner with J. Harer, Combinatorics of train tracks, Ann. Math. Stud. 125, Princeton Univ. Press, Princeton, 1992 MR 1144770 (94b:57018)

21.
U. Oertel, Homology branched surfaces: Thurston's norm on $ H_2(M^3)$, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), 253-272, London Math. Soc. Lecture Notes Ser. 112, Cambridge Univ. Press, Cambridge, 1986 MR 903869 (89e:57011)

22.
R. Rannard, Computing immersed normal surfaces in the figure-eight knot complement, Experiment. Math. 8 (1999), no. 1, 73-84 MR 1685039 (2000h:57031)

23.
P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555-565 MR 0494062 (58:12996)

24.
J.-P. Serre, Trees, Springer Monographs in Mathematics (corrected 2nd printing, trans. J. Stillwell), Springer-Verlag, Berlin, 2003 MR 1954121 (2003m:20032)

25.
W. Thurston, A norm for the homology of $ 3$-manifolds, Mem. Amer. Math. Soc. 59 (1986), no. 339, i-vi and 99-130 MR 823443 (88h:57014)

26.
D. Zhuang, Irrational stable commutator length in finitely presented groups, Jour. Mod. Dyn. 2 (2008) no. 3, 499-507 MR 2417483

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Additional Information:

Danny Calegari
Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125
Email: dannyc@its.caltech.edu

DOI: 10.1090/S0894-0347-09-00634-1
PII: S 0894-0347(09)00634-1
Received by editor(s): February 18, 2008
Posted: May 1, 2009
Dedicated: Dedicated to Shigenori Matsumoto on the occasion of his 60th birthday
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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