# Groups, Geometry, and Dynamics

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**Volume 4, Issue 1, 2010, pp. 15–29**

**DOI: 10.4171/GGD/73**

Published online: 2009-12-23

Free subalgebras of Lie algebras close to nilpotent

Alexey Belov^{[1]}and Roman Mikhailov

^{[2]}(1) Bar-Ilan University, Ramat Gan, Israel

(2) Steklov Mathematical Institute, Moscow, Russian Federation

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`G`_{n + 2}
be a group with `n` + 2`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.

*Keywords: *Lie algebra, automata algebra, free group, nilpotency

Belov Alexey, Mikhailov Roman: Free subalgebras of Lie algebras close to nilpotent. *Groups Geom. Dyn.* 4 (2010), 15-29. doi: 10.4171/GGD/73