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 ≥ 1, let Ln + 2 be a Lie algebra with generators x1, …, xn + 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 {x1, …, xn + 2} is zero. As an application of this result about automata algebras, we prove that Ln + 2 contains a free subalgebra for every n ≥ 1. We also prove the similar result about groups defined by commutator relations. Let Gn + 2 be a group with n + 2 generators y1, …, yn + 2 and the following relations: for k ≤ n, any left-normalized commutator of length k which consists of fewer than k different symbols from {y1, …, yn + 2} is trivial. Then the group Gn + 2 contains a 2-generated free subgroup.

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