Generalizing Magnus' characterization of free groups to some free\n products

A residually nilpotent group is \\emph{$k$-parafree} if all of its lower\ncentral series quotients match those of a free group of rank $k$. Magnus proved\nthat $k$-parafree groups of rank $k$ are themselves free. In this note we mimic\nthis theory with finite extensions of free groups, with an emphasis on free\nproducts of the cyclic group $C_p$, for $p$ an odd prime. We show that for $n\n\\leq p$ Magnus' characterization holds for the $n$-fold free product $C_p^{*n}$\nwithin the class of finite-extensions of free groups. Specifically, if $n \\leq\np$ and $G$ is a finitely generated, virtually free, residually nilpotent group\nhaving the same lower central series quotients as $C_p^{*n}$, then $G \\cong\nC_p^{*n}$. We also show that such a characterization does not hold in the class\nof finitely generated groups. That is, we construct a rank 2 residually\nnilpotent group $G$ that shares all its lower central series quotients with\n$\\ffp$, but is not $\\ffp$.\n