The Formanek-Procesi group with base a right-angled Artin group: Residual nilpotence and Lie algebra

Abstract

In the present work, we investigate the Lie algebra of the Formanek-Procesi group F P (H) with base group H a right-angled Artin group. We show that the Lie algebra gr(FP(H)) has a presentation that is dictated by the group presentation. As a result, we are able to show that F P (H) is residually nilpotent.

Lower central series and split extensions

“…By Theorem 8.1, though, this kernel is the Lie algebra gr L pAq, which indeed has degree 1 piece equal to Z " xa 3 y, but also has degree 2 piece equal to Z 2 " xra 1 , a 3 s, ra 2 , a 3 sy, and so on. In fact, it follows from [32,25] that gr L n pAq -Lie n´1 pZ 2 q for n ě 2, although of course grpAq " LiepZ 3 q.…”