On quasi-homomorphisms and commutators in the special linear group over a euclidean ring

We prove that for any euclidean ring R and n ≥ 6, Γ = SLn(R) has no unbounded quasi-homomorphisms. By Bavard's duality theorem, this means that the stable commutator length vanishes on Γ. The result is particularly interesting for R = F[x] for a certain field F (such as ), because in this case the commutator length on Γ is known to be unbounded. This answers a question of M. Abért and N. Monod for n ≥ 6.