## Stable commutator length is rational in free groups

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- by Danny Calegari
- J. Amer. Math. Soc.
**22**(2009), 941-961 - DOI: https://doi.org/10.1090/S0894-0347-09-00634-1
- Published electronically: 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.

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## Bibliographic Information

**Danny Calegari**- Affiliation: Department of Mathematics, Caltech, Pasadena, California 91125
- MR Author ID: 605373
- Email: dannyc@its.caltech.edu
- Received by editor(s): February 18, 2008
- Published electronically: May 1, 2009
- © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**22**(2009), 941-961 - MSC (2000): Primary 57M07, 20F65, 20J05
- DOI: https://doi.org/10.1090/S0894-0347-09-00634-1
- MathSciNet review: 2525776

Dedicated: Dedicated to Shigenori Matsumoto on the occasion of his 60th birthday