On semiconcise words

Let $w$ be a group-word. For a group $G$, let $G_w$ denote the set of all $w$-values in $G$ and $w(G)$ the verbal subgroup of $G$ corresponding to $w$. The word $w$ is semiconcise if the subgroup $[w(G),G]$ is finite whenever $G_w$ is finite. The group $G$ is an $FC(w)$-group if the set of conjugates $x^{G_w}$ is finite for all $xin G$. We prove that if $w$ is a semiconcise word and $G$ is an $FC(w)$-group, then the subgroup $[w(G),G]$ is $FC$-embedded in $G$, that is, the intersection $C_G(x)cap [w(G),G]$ has finite index in $[w(G),G]$ for all $xin G$. A similar result holds for $BFC(w)$-groups, that are groups in which the sets $x^{G_w}$ are boundedly finite. We also show that this is no longer true if $w$ is not semiconcise.