15.1 (2002)

(P. Longobardi, M. Maj, A. H. Rhemtulla). Let $w = w(x_1, \dots, x_n)$ be a group word in $n$ variables $x_1, \dots, x_n$, and $V(w)$ the variety of groups defined by the law $w = 1$. Let $V(w^*)$ (respectively, $V(w^\#)$) be the class of all groups $G$ in which for every $n$ infinite subsets $S_1, \dots, S_n$ there exist $s_i \in S_i$ such that $w(s_1, \dots, s_n) = 1$ (respectively, $\langle s_1, \dots, s_n \rangle \in V(w)$).
$\qquad$ a) Is there some word $w$ and an infinite group $G$ such that $G \in V(w^\#)$ but $G \notin V(w)$?
$\qquad$ b) Is there some word $w$ and an infinite group $G$ such that $G \in V(w^*)$ but $G \notin V(w^\#)$?
The answer to both of these questions is likely to be "yes". It is known that in a) $w$ cannot be any of several words such as $x_1^n$, $[x_1, \dots, x_n]$, $[x_1, x_2]^2$, $(x_1x_2)^3 x_2^{-3} x_1^{-3}$, and $x_1^{a_1} \dots x_n^{a_n}$ for any non-zero integers $a_1, \dots, a_n$.