Generalizing Magnus' characterization of free groups to some free products

TL;DR

This paper extends Magnus' characterization of free groups to certain free products, showing that under specific conditions, groups with matching lower central series quotients are isomorphic to these free products.

Contribution

It generalizes Magnus' theorem to free products of cyclic groups, identifying conditions under which such groups are uniquely characterized by their lower central series.

Findings

Magnus' characterization holds for free products of cyclic groups when n ≤ p.

Counterexamples show the characterization fails in the broader class of finitely generated groups.

The paper constructs a residually nilpotent group sharing quotients with a free product but not isomorphic to it.

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 $\ffp$.