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The free multiplicative Lie algebra L(P) for a finitely generated parafree group P
Abstract
Let P be a group, L(P) be the free multiplicative Lie algebra with normal subgroup Γn(P) generated by Lie bracket ``commutators'' of weight n and { γn(P) } be the lower central series of P. We prove that Γn(P) ≃ γn(P) for arbitrary n ≥ 1 and P a finitely generated parafree group such that H2(P, ℤ) = 0 = H3(P, ℤ) (e.g. P satisfies the Strong Parafree Conjecture), in particular Ellis's conjecture holds i.e. the above isomorphism holds for finitely generated free group P but this easily implies it holds for any free group.
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BibTeX
@misc{kochloukova2026free,
title = {{The free multiplicative Lie algebra $L(P)$ for a finitely generated parafree group $P$}},
author = {Dessislava H. Kochloukova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19595},
eprint = {2609.19595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19595},
}
RIS
TY - UNPB
TI - The free multiplicative Lie algebra L(P) for a finitely generated parafree group P
AU - Dessislava H. Kochloukova
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19595
UR - https://arxiv.org/abs/2609.19595
AB - Let P be a group, L(P) be the free multiplicative Lie algebra with normal subgroup Γ_n(P) generated by Lie bracket ``commutators'' of weight n and { γ_n(P) } be the lower central series of P. We prove that Γ_n(P) ≃ γ_n(P) for arbitrary n ≥ 1 and P a finitely generated parafree group such that H_2(P, ℤ) = 0 = H_3(P, ℤ) (e.g. P satisfies the Strong Parafree Conjecture), in particular Ellis's conjecture holds i.e. the above isomorphism holds for finitely generated free group P but this easily implies it holds for any free group.
ER -