Commutators of trace zero matrices over principal ideal rings

TL;DR

This paper proves that every trace zero matrix over a principal ideal ring is a commutator of two trace zero matrices, with the added condition that one can be regular mod every maximal ideal, and confirms Shalev's conjecture for 2x2 matrices over p-adic integers.

Contribution

It establishes that trace zero matrices over principal ideal rings are commutators of trace zero matrices, simplifying previous proofs and confirming a conjecture for 2x2 matrices over p-adic integers.

Findings

Every trace zero matrix over a principal ideal ring is a commutator of trace zero matrices.

One of the matrices can be chosen to be regular mod every maximal ideal.

Shalev's conjecture is proven for 2x2 matrices over p-adic integers.

Abstract

We prove that for every trace zero matrix $A$ over a principal ideal ring $R$, there exist trace zero matrices $X,Y$ over $R$ such that $XY-YX=A$. Moreover, we show that $X$ can be taken to be regular mod every maximal ideal of $R$. This strengthens our earlier result that $A$ is a commutator of two matrices (not necessarily of trace zero), and in addition, the present proof is significantly simpler than the earlier one. Shalev has conjectured an analogous statement for group commutators in $\mathrm{SL}_{n}$ over $p$-adic integers. We prove Shalev's conjecture for $n=2$.