Ellipticity of words (Q1702700)

Let \(G\) be a pro-\(p\) group and \(w=w(x_1, \ldots ,x_n)\in F_n\) an abstract word on \(n\) letters. It is said that \(w\) is elliptic (or has finite verbal width) on \(G\) if \(\langle \overline{w(G)}\rangle =w(G)^{\pm 1}\stackrel{n)}{\cdots} w(G)^{\pm 1}\) for some \(n\geqslant 1\), where \(w(G)=\{ w(g_1, \ldots ,g_n) \mid g_i\in G\}\subseteq G\) is the set of values of \(w\) in \(G\), and \(\langle \overline{w(G)}\rangle\) is the closure of the subgroup of \(G\) generated by \(w(G)\). Equivalently, \(w\) is elliptic if and only if the discrete subgroup of \(G\) generated by \(w(G)\) is closed. It is known that, on some important classes of groups (like \(p\)-analytic pro-\(p\) groups, the Notingham group, etc), all words are elliptic. The paper under review contributes in this direction proving that all pro-\(p\) completions of finitely generated residually-\(p\) torsion groups also belong to this family: all words are elliptic. Further, the author also proves that in such groups, any multilinear word (see the technical definition in the paper) is strongly elliptic (a somehow stronger condition of ellipticity).