JournalsggdVol. 4 , No. 1DOI 10.4171/ggd/73

Free subalgebras of Lie algebras close to nilpotent

  • Alexey Belov

    Bar-Ilan University, Ramat Gan, Israel
  • Roman Mikhailov

    Steklov Mathematical Institute, Moscow, Russian Federation
Free subalgebras of Lie algebras close to nilpotent cover

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.