# Free subalgebras of Lie algebras close to nilpotent

### Alexey Belov

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

Steklov Mathematical Institute, Moscow, Russian Federation

## Abstract

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.

## Cite this article

Alexey Belov, Roman Mikhailov, Free subalgebras of Lie algebras close to nilpotent. Groups Geom. Dyn. 4 (2010), no. 1, pp. 15–29

DOI 10.4171/GGD/73