Title: Generalizing Magnus' Characterization of Free Groups to Some Free Products
Abstract: Abstract A residually nilpotent group is 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 ≤ p Magnus’ characterization holds for the n-fold free product within the class of finite-extensions of free groups. Specifically, if n ≤ p and G is a finitely generated, virtually free, residually nilpotent group having the same lower central series quotients as , then . 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 C p *C p , but is not C p *C p . Key Words: Free productsLower central seriesParafreeResidually nilpotent2010 Mathematics Subject Classification: 20E2620E18 ACKNOWLEDGMENT Finally, we are thankful to Tom Church for comments on a previous draft of this paper. Notes Communicated by P. Tiep.