IA-automorphisms of relatively free nilpotent torsion-free groups and Lie algebras

Abstract

For a free metabelian and nilpotent group G of finite rank n and class c, with $n\ge 2$ and $c\ge 3,$ we show that ${\gamma }_r({\text{IA}}_G)\ne {\mathrm{I}}_{r+1}{\mathrm{A}}_G$ for all $2\le r\le c-1$ and we extend this result to several torsion-free nilpotent varieties of groups. By our methods, which are based on a connection of automorphisms of groups with automorphisms of Lie algebras, we also show the analogous result for several relatively free nilpotent Lie algebras.

Keywords:

• Free metabelian and nilpotent groups and Lie algebras
• IA-automorphisms
• Mal’cev Lie algebra

MATHEMATICS SUBJECT CLASSIFICATION 2010:

• 20F18
• 20F28
• 17B01
• 17B30
• 17B40
• 20E10