17.109 (2010)

A non-trivial group word $w$ is uniformly elliptic in a class $\mathcal{C}$ if there is a function $f: \mathbb{N} \to \mathbb{N}$ such that the width of $w$ in every $d$-generator $\mathcal{C}$-group $G$ is bounded by $f(d)$ (i.e. every element of the verbal subgroup $w(G)$ is equal to a product of $f(d)$ values of $w$ or their inverses). If $\mathcal{C}$ is a class of finite groups, this is equivalent to saying that in every finitely generated pro-$\mathcal{C}$ group $G$ the verbal subgroup $w(G)$ is closed. A. Jaikin-Zapirain (Revista Mat. Iberoamericana, 24 (2008), 617–630) proved that $w$ is uniformly elliptic in finite $p$-groups if and only if $w \notin F''(F')^p$, where $F$ is the free group on the variables of $w$. Is it true that $w$ is uniformly elliptic in $\mathcal{C}$ if and only if $w \notin F''(F')^p$ for every prime $p$ in the case where $\mathcal{C}$ is the class of all
$\qquad$ a) finite soluble groups?
$\qquad$ b) finite groups?

See also Ch. 4 of (D. Segal, Words: notes on verbal width in groups, LMS Lect. Note Series, 361, Cambridge Univ. Press, 2009.