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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
DOI: https://doi.org/10.1090/S0894-0347-09-00634-1
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.


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Additional Information

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

DOI: https://doi.org/10.1090/S0894-0347-09-00634-1
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.

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