Free subalgebras of Lie algebras close to nilpotent

Abstract

We prove that for every automata algebra of exponential growth the associated Lie algebra contains a free subalgebra. For $n\ge 1$, let $L_{n+2}$ be a Lie algebra with generators $x_1,\dots ,x_{n+2}$ and the following relations: for $k\le n$, any commutator (with any arrangement of brackets) of length $k$ which consists of fewer than $k$ different symbols from $\{x_1,\dots ,x_{n+2}\}$ is zero. As an application of this result about automata algebras, we prove that $L_{n+2}$ contains a free subalgebra for every $n\ge 1$. We also prove the similar result about groups defined by commutator relations. Let $G_{n+2}$ be a group with $n+2$ generators $y_1,\dots ,y_{n+2}$ and the following relations: for $k\le n$, any left-normalized commutator of length $k$ which consists of fewer than $k$ different symbols from $\{y_1,\dots ,y_{n+2}\}$ is trivial. Then the group $G_{n+2}$ contains a 2-generated free subgroup.

The main technical tool is combinatorics of words, namely combinatorics of periodical sequences and period switching.