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.
1
Bar-Ilan University, Ramat Gan, Israel
2
Steklov Mathematical Institute, Moscow, Russian Federation
Alexey Belov; Roman Mikhailov. Free subalgebras of Lie algebras close to nilpotent. Groups, geometry, and dynamics, Tome 4 (2010) no. 1, pp. 15-29. doi: 10.4171/ggd/73
@article{10_4171_ggd_73,
author = {Alexey Belov and Roman Mikhailov},
title = {Free subalgebras of {Lie} algebras close to nilpotent},
journal = {Groups, geometry, and dynamics},
pages = {15--29},
year = {2010},
volume = {4},
number = {1},
doi = {10.4171/ggd/73},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/73/}
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AU - Roman Mikhailov
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