On the verbal width of finitely generated pro-$p$ groups

Abstract

Let $p$ be a prime. It is proved that a non-trivial word $w$ from a free group $F$ has finite width in every finitely generated pro-$p$ group if and only if $w\notin (F^{\prime }{)}^p{F}^{\prime \prime }$. Also it is shown that any word $w$ has finite width in a compact $p$-adic group.