Generalising a Theorem of Lichtman

Abstract

We show that under a suitable additional hypothesis the restricted Zassenhaus $\F_p$-Lie algebra or the rational Magnus Lie algebra of a free amalgamated product is the free amalgamated product of the corresponding Lie algebras of the factors. This generalises a Theorem of A.I.\,Lichtman, who proved the analoguous statement for free products. Our conditions include the case when the amalgamated product is a retract in both factors. As a by-product, we show that a free product of residually torsion free nilpotent groups amalgamated along retracts is also residually torsion free nilpotent and obtain also some results on cohomological completeness. In the final sections we apply our main results to two recently raised open questions.