Lie algebras and torsion groups with identity

TL;DR

This paper proves that certain finitely generated Lie algebras with specific nilpotency conditions are actually nilpotent, leading to a conclusion that related residually-$p$ torsion groups are finite.

Contribution

It establishes a new nilpotency criterion for finitely generated Lie algebras satisfying polynomial identities and ad-nilpotency conditions.

Findings

Finitely generated Lie algebras with ad-nilpotent commutators and polynomial identities are nilpotent.

Residually-$p$ torsion groups with pro-$p$ identities are finite.

Provides a link between Lie algebra properties and group finiteness.

Abstract

We prove that a finitely generated Lie algebra $L$ such that (i) every commutator in generators is ad-nilpotent, and (ii) $ L$ satisfies a polynomial identity, is nilpotent. As a corollary we get that a finitely generated residually-$p$ torsion group whose pro-$p$ completion satisfies a pro-$p$ identity is finite.