Geodesic
Stable commutator length is rational in free groups
Calegari, Danny1
1Department of Mathematics, Caltech, Pasadena, California 91125
Journal of the American Mathematical Society, Tome 22 (2009) no. 4, pp. 941-961

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-09-00634-1

Calegari, Danny  1

1 Department of Mathematics, Caltech, Pasadena, California 91125 
Calegari, Danny. Stable commutator length is rational in free groups. Journal of the American Mathematical Society, Tome 22 (2009) no. 4, pp. 941-961. doi: 10.1090/S0894-0347-09-00634-1
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