Expressing Finite-Infinite Matrices Into Products of Commutators of Finite Order Elements

TL;DR

This paper demonstrates that elements of certain infinite matrix groups over rings can be expressed as products of a bounded number of commutators of elements with finite order, extending classical results to infinite-dimensional settings.

Contribution

It provides explicit bounds and constructions for expressing elements as products of commutators of finite order elements in infinite matrix groups.

Findings

Every element of $UT_{ ext{infinity}}(R)$ is a product of $4k-6$ commutators of elements of order $k$.

In $SL_n(R)$ and $SL_{VK}( ext{infinity}, R)$ over $ ext{Re}$ or $ ext{C}$, elements can be decomposed into at most $4k-6$ commutators.

The results extend finite-dimensional commutator decompositions to infinite-dimensional matrix groups.

Abstract

Let $R$ be an associative ring with unity $1$ and consider $k\in \mathbb{N}$ such that $1+1+..+1=k$ is invertible. Denote by $\omega$ an arbitrary kth root of unity in $R$ and let $UT^{(k)}_{\infty}(R)$ be the group of upper triangular infinite matrices whose diagonal entries are $k$th roots of $1$. We show that every element of the group $UT_{\infty}(R)$ can be expressed as a product of $4k-6$ commutators all depending of powers of elements in $UT^{(k)}_{\infty}(R)$ of order $k$. If $R$ is the complex field or the real number field we prove that, in $SL_n(R)$ and in the subgroup $SL_{VK}(\infty,R)$ of the Vershik-Kerov group over $R$, each element in these groups can be decomposed into a product of at most $4k-6$ commutators of elements of order $k$.