PreprintCC BY 4.0arXiv
Solvability of unimodular equations in groups and Lie algebras
Abstract
Our results imply, in particular, that a finitely generated solvable group G is nilpotent if and only if it contains a solution to any unimodular equation, i.e., an equation of the form ∏ gixni=1, where gi∈ G and ∑ ni=±1. A similar fact turns out to be true for Lie algebras. We also exhibit an example of a unimodular equation w(x)=g over a finitely generated group G, which has a solution (in G) for any g∈ G, but the solution is not unique for some g∈ G. We show that, for nilpotent groups G, the set of unimodular mappings Gn→ Gn (which are defined naturally) forms a group under the composition.
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BibTeX
@misc{klyachko2026solvability,
title = {{Solvability of unimodular equations in groups and Lie algebras}},
author = {Anton A. Klyachko and Mikhail A. Mikheenko and Alexander Yu. Olshanskii},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.28045},
eprint = {2608.28045},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28045},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Solvability of unimodular equations in groups and Lie algebras
AU - Anton A. Klyachko
AU - Mikhail A. Mikheenko
AU - Alexander Yu. Olshanskii
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2608.28045
UR - https://arxiv.org/abs/2608.28045
AB - Our results imply, in particular, that a finitely generated solvable group G is nilpotent if and only if it contains a solution to any unimodular equation, i.e., an equation of the form ∏ g_ix^n_i=1, where g_i∈ G and ∑ n_i=±1. A similar fact turns out to be true for Lie algebras. We also exhibit an example of a unimodular equation w(x)=g over a finitely generated group G, which has a solution (in G) for any g∈ G, but the solution is not unique for some g∈ G. We show that, for nilpotent groups G, the set of unimodular mappings G^n→ G^n (which are defined naturally) forms a group under the composition.
ER -