Generalizing Magnus' Characterization of Free Groups to Some Free Products

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 products
• Lower central series
• Parafree
• Residually nilpotent

2010 Mathematics Subject Classification:

• 20E26
• 20E18

ACKNOWLEDGMENT

Finally, we are thankful to Tom Church for comments on a previous draft of this paper.

Notes

Communicated by P. Tiep.