On Quasi-homomorphisms and Commutators in the Special Linear Group over a Euclidean Ring
Group Theory
2010-09-24 v4 Geometric Topology
Abstract
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.
Keywords
Cite
@article{arxiv.0911.1341,
title = {On Quasi-homomorphisms and Commutators in the Special Linear Group over a Euclidean Ring},
author = {Masato Mimura},
journal= {arXiv preprint arXiv:0911.1341},
year = {2010}
}
Comments
This is the final version. 8 pages; title changed again; title changed, a little generalization of the main theorem