Stable commutator length on mapping class groups

Let Γ be a finite index subgroup of the mapping class group M C G ( § ) of a closed orientable surface § , possibly with punctures. We give a precise condition (in terms of the Nielsen-Thurston decomposition) when an element g ∈ Γ 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 s c l is uniformly positive on the nontrivial elements of the Torelli group. The proofs use our previous construction of group actions on quasi-trees.