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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

MR Author ID:
605373

Email:
dannyc@its.caltech.edu

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.