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- W4301174166 abstract "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 $n leq p$ Magnus' characterization holds for the $n$-fold free product $C_p^{*n}$ within the class of finite-extensions of free groups. Specifically, if $n leq p$ and $G$ is a finitely generated, virtually free, residually nilpotent group having the same lower central series quotients as $C_p^{*n}$, 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 $ffp$, but is not $ffp$." @default.
- W4301174166 created "2022-10-04" @default.
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- W4301174166 date "2010-04-01" @default.
- W4301174166 modified "2023-09-29" @default.
- W4301174166 title "Generalizing Magnus' characterization of free groups to some free products" @default.
- W4301174166 doi "https://doi.org/10.48550/arxiv.1004.0222" @default.
- W4301174166 hasPublicationYear "2010" @default.
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