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

TL;DR

This paper proves that for Euclidean rings and matrices of size at least 6, the special linear group has no unbounded quasi-homomorphisms, implying the stable commutator length is zero, contrasting with known unbounded cases.

Contribution

It establishes the absence of unbounded quasi-homomorphisms in SL_n(R) for Euclidean rings R and n ≥ 6, answering a question about commutator lengths.

Findings

No unbounded quasi-homomorphisms in SL_n(R) for Euclidean rings R and n ≥ 6

Stable commutator length vanishes on these groups

Contrasts with unbounded commutator length in certain polynomial rings

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.