## 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.

Original language | English |
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Pages (from-to) | 15-29 |

Number of pages | 15 |

Journal | Groups, Geometry, and Dynamics |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - 2010 |

## Keywords

- Automata algebra
- Free group
- Lie algebra
- Nilpotency