Free Subalgebras of Lie Algebras Close to Nilpotent

We prove that for every automata algebra of exponential growth, the associated Lie algebra contains a free subalgebra. For n ≥ 1, let Ln+2 be a Lie algebra with generator set x1, . . . , xn+2 and the following relations: for k ≤ n, any commutator of length k which consists of fewer than k different symbols from {x1, . . . , xn+2} is zero. As an application of this result about automata algebras, we prove that for every n ≥ 1, Ln+2 contains a free subalgebra. We also prove the similar result about groups defined by commutator relations.