Geodesic
Stable commutator length is rational in free groups
Calegari, Danny1
1 Department 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 
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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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