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 L _n+2 be a Lie algebra with generators x_1,⋯, x _n+2 and the following relations: for k ≤ n, any commutator (with any arrangement of brackets) of length k which consists of fewer than k different symbols from {x_1,⋯, 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 ≥ 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,⋯, y_n+2 and the following relations: for k ≤ n, any left-normalized commutator of length k which consists of fewer than k different symbols from {y_1,⋯, 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.