Multiple commutators of elementary subgroups: end of the line

TL;DR

This paper proves multiple commutator formulas for elementary subgroups in $GL(n,R)$ over any associative ring, removing previous restrictions on the ring's properties and advancing the understanding of subgroup interactions.

Contribution

It introduces a new approach that lifts previous assumptions, establishing comprehensive multiple commutator formulas for elementary subgroups over arbitrary rings.

Findings

Multiple commutator formulas are valid over any associative ring.

Reduces complex multiple commutators to simpler double commutators.

Extends previous results to a broader class of rings.

Abstract

In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we established commutator formulas for relative elementary subgroups in $GL(n,R)$, $n\ge 3$, and other similar groups, such as Bak's unitary groups, or Chevalley groups. In particular, there it was shown that multiple commutators of elementary subgroups can be reduced to double such commutators. However, since the proofs of these results depended on the standard commutator formulas, it was assumed that the ground ring $R$ is quasi-finite. Here we propose a different approach which allows to lift any such assumptions and establish almost definitive results. In particular, we prove multiple commutator formulas, and other related facts for $GL(n,R)$ over an {\it arbitrary} associative ring $R$.