Free subalgebras of Lie algebras close to nilpotent

What is it about?

For a positive integer $n$, let $L_{n+2}$ be a Lie algebra with generators $x_{1}, \dots, x_{n+2}$ and the following relations: for $k \leq n$, any commutator (with any arrangement of brackets) of length $k$ that consists of fewer than $k$ different symbols from $\{x_{1}, \dots, x_{n+2}\}$ is zero. In the paper under review, it is shown that for every automata algebra of exponential growth the associated Lie algebra contains a free subalgebra. As an application, it is proved that $L_{n+2}$ contains a free subalgebra for every $n \geq 1$. Its proof is based on the theory of monomial algebras. A similar result is obtained about groups. Reviewed by Athanassios I. Papistas

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