Stable commutator length on mapping class groups

Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji

Annales de l'Institut Fourier, Tome 66 (2016) no. 3, pp. 871-898.

Soit $\Gamma$ un sous-groupe d’indice fini du groupe modulaire $MCG\left(\S \right)$ d’une surface fermée orientable, possiblement épointée. Nous donnons une condition précise (en termes de la décomposition de Nielsen-Thurston) pour qu’un élément $g\in \Gamma$ ait une longueur stable des commutateurs strictement positive. Nous montrons de plus que dans ces situations, la longueur stable des commutateurs est soit nulle, soit uniformément minorée par un réel strictement positif. Notre méthode permet aussi de traiter le cas de certains sous-groupes d’indice infini, et nous montrons l’existence d’un minorant strictement positif pour la longueur stable des commutateurs des éléments non triviaux du groupe de Torelli. Les démonstrations utilisent notre prééédente construction d’actions de groupes sur des quasi-arbres.

Let $\Gamma$ be a finite index subgroup of the mapping class group $MCG\left(\S \right)$ of a closed orientable surface $\S$, possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element $g\in \Gamma$ has positive stable commutator length. In addition, we show that in these situations the stable commutator length, if nonzero, is uniformly bounded away from 0. The method works for certain subgroups of infinite index as well and we show $scl$ is uniformly positive on the nontrivial elements of the Torelli group. The proofs use our previous construction of group actions on quasi-trees.

Classification : 20F65
Keywords: stable commutator length, mapping class groups, quasi-morphisms, projection complex
Mot clés : Longueur stable des commutateurs

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