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On the residual finiteness of the non-abelian tensor square G ⊗ G, the non-abelian exterior square G ∧ G and the weak commutativity construction X(G)
Abstract
We prove that if G is a residually finite group, Ĝ is its profinite completion and the map $H_2(G, \mathbb{Z}) $ $ \to H_2(\widehat{G}, \widehat{\mathbb{Z}}),$ induced by the canonical map G → Ĝ, is injective, then the non-abelian exterior square G ∧ G is residually finite. We show that if G is a finitely presented centre-by-metabelian group then ν(G), the non-abelian tensor product G ⊗ G and the non-abelian tensor square G ∧ G are residually finite. Furthermore we prove that if G be a finitely presented metabelian group then the weak commutative construction X(G) is residually finite. We discuss a criterion for the q-exterior square G ∧q G to be residually finite.
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BibTeX
@misc{teixeira2026residual,
title = {{On the residual finiteness of the non-abelian tensor square $G \otimes G$, the non-abelian exterior square $G \wedge G$ and the weak commutativity construction $\X(G)$}},
author = {Ayrton Anjos Teixeira and Lucas Barroso Rocha and Dessislava H. Kochloukova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34057},
eprint = {2609.34057},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34057},
}
RIS
TY - UNPB
TI - On the residual finiteness of the non-abelian tensor square G ⊗ G, the non-abelian exterior square G ∧ G and the weak commutativity construction X(G)
AU - Ayrton Anjos Teixeira
AU - Lucas Barroso Rocha
AU - Dessislava H. Kochloukova
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34057
UR - https://arxiv.org/abs/2609.34057
AB - We prove that if G is a residually finite group, Ĝ is its profinite completion and the map $H_2(G, \mathbb{Z}) $ $ \to H_2(\widehat{G}, \widehat{\mathbb{Z}}),$ induced by the canonical map G → Ĝ, is injective, then the non-abelian exterior square G ∧ G is residually finite. We show that if G is a finitely presented centre-by-metabelian group then ν(G), the non-abelian tensor product G ⊗ G and the non-abelian tensor square G ∧ G are residually finite. Furthermore we prove that if G be a finitely presented metabelian group then the weak commutative construction X(G) is residually finite. We discuss a criterion for the q-exterior square G ∧^q G to be residually finite.
ER -