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(R) 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.Comment: This is the final version. 8 pages; title changed again; title changed, a little generalization of the main theore

Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity

Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.Comment: 50 page