Baer invariants and residual nilpotence of groups

We study descending chains of subgroups in the Baer invariants, which naturally generalize the Dwyer filtration of the multiplicator of a group. We establish a connection between these structures and residual nilpotence of groups. As an application of our methods, we construct a finitely presented residually nilpotent group $F/R$ none of whose free $k$-central extensions $F/[R,_kF]$ ($k\geqslant 1$) is residually nilpotent. For $k=1,2$, it is shown that the residual nilpotence of a free product $G$ of one-relator groups is equivalent to the residual nilpotence of any $k$-central extension of $G$.