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

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

2009-11-06

arxiv.org/abs/0911.1341v4

Int. Math. Res. Not. IMRN, 2010 no. 18, 3519--3529

Mathematics

Group Theory

This is the final version. 8 pages; title changed again; title changed, a little generalization of the main theorem

Scientific paper

10.1093/imrn/rnq011

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.

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