On Quasi-homomorphisms and Commutators in the Special Linear Group over a Euclidean Ring

We prove that for any euclidean ring R and n at least 6, Gamma=SL_n® has no unbounded quasi-homomorphisms. From Bavard’s duality theorem, this means that the stable commutator length vanishes on Gamma. The result is particularly interesting for R = F[x] for a certain field F (such as the field C of complex numbers, because in this case the commutator length on Gamma is known to be unbounded. This answers a question of M. Ab'ert and N. Monod for n at least 6.