12.95 (1992)

Let $G$ be a finitely generated pro-$p$-group and let $g_1, \dots, g_n \in G$. Let $H$ be an open subgroup of $G$, and suppose there exists a non-trivial word $w = w(X_1, \dots, X_n)$ such that $w(a_1, \dots, a_n) = 1$ whenever $a_1 \in g_1 H, \dots, a_n \in g_n H$ (that is, $G$ satisfies a coset identity). Does it follow that $G$ satisfies some non-trivial identity?

A positive answer to this question would imply that an analogue of the Tits Alternative holds for finitely generated pro-$p$-groups. Note that J. S. Wilson and E. I. Zelmanov (J. Pure Appl. Algebra, 81 (1992), 103–109) showed that the graded Lie algebra $L_p(G)$ with respect to the dimension subgroups of $G$ in characteristic $p$ satisfies a polynomial identity.