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

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 $U{T}_{\infty }^{(k)}(R)$ 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 $U{T}_{\infty }(R)$ can be expressed as a product of $4k-6$ commutators all depending of powers of elements in $U{T}_{\infty }^{(k)}(R)$ of order $k$. If $R$ is the complex field or the real number field we prove that, in $S{L}_n(R)$ and in the subgroup $S{L}_{VK}(\infty ,R)$ 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 $k$.

10 pages

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