Generalizing Magnus' characterization of free groups to some free products

A residually nilpotent group is \emph{$k$-parafree} if all of its lower central series quotients match those of a free group of rank $k$. Magnus proved that $k$-parafree groups of rank $k$ are themselves free. In this note we mimic this theory with finite extensions of free groups, with an emphasis on free products of the cyclic group $C_p$, for $p$ an odd prime. We show that for $n \leq p$Magnu{s}^{\mathrm{\prime }}characterizationholdsforthe$n$-foldfreeproduct$C_p^{*n}$withintheclassoffinite-extensionsoffreegroups\mathrm{.}Specifically,if$n \leq p$and$G$ is a finitely generated, virtually free, residually nilpotent group having the same lower central series quotients as $C_p^{\ast n}$, then $G \cong C_p^{*n}$. We also show that such a characterization does not hold in the class of finitely generated groups. That is, we construct a rank 2 residually nilpotent group $G$ that shares all its lower central series quotients with $\text{ffp}$, but is not $\text{ffp}$.