On the Width of Verbal Subgroups of the Groups of Triangular Matrices over a Field of Arbitrary Characteristic

Abstract

The width $\wid(G,W)$ of the verbal subgroup $v(G,W)$ of a group $G$ defined by a collection of group words $W$ is the smallest number $m$ in $\mathbb N \cup {+\infty}$ such that every element of $v(G,W)$ is can be represented as the product of at most $m$ words in $W$ evaluated on the group $G$ and their inverses. Recall that every verbal subgroup of the group $T_{n} (K)$ of triangular matrices over an arbitrary field $K$ can be defined by just one word: an outer commutator word or a power word. We prove that for every outer commutator word $w$ the equality $\wid(T_{n} (K),w)=1$ holds on the group $T_{n} (K)$ and that if $w=x^{s}$ then $\wid(T_{n}(K),w)=1$ except in two cases: (1) the field $K$ is finite and $s$ is divisible by the characteristic $p$ of $K$ but not by $|K|-1$; (2) the field $K$ is finite and $s=p^{t} (|K|-1)^{u} r$ for $r,t,u\in \mathbb N$ with $n\ge p^{t} +3$, while $r$ not divisible by $p$. In these cases the width equals 2. For finitary triangular groups the situation is similar, but in the second case the restriction $n\ge p^{t} +3$ is superfluous.