17.81 (2010)

Given normal subgroups $R_1, \dots, R_n$ of a group $G$, let $[[R_1, \dots, R_n]] := \prod ( [\bigcap_{i \in I} R_i, \bigcap_{j \in J} R_j] )$, where the product is over all $I \cup J = \{1, \dots, n\}$, $I \cap J = \varnothing$.

Let $G$ be a free group, and let $R_i = \langle r_i \rangle^G$ be the normal closures of elements $r_i \in G$. It is known (B. Hartley, Yu. Kuzmin, J. Pure Appl. Algebra, 74 (1991), 247–256) that the quotient $(R_1 \cap R_2)/[R_1, R_2]$ is a free abelian group. On the other hand, the quotient $(R_1 \cap \dots \cap R_n)/[[R_1, \dots, R_n]]$ has, in general, non-trivial torsion for $n \geqslant 4$. Is this quotient always torsion-free for $n = 3$? This is known to be true if $r_1, r_2, r_3$ are not proper powers in $G$.