On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and SLn(Z)
Abstract
We show that cohomological and homological goodness are equivalent. We prove that if G is a coherent even Artin group or an even Artin group whose underlying graph does not contain 4-clique or a LERF Artin group then G ⊗ G and G ∧ G are residually finite. Furthermore for G a RAAG both G ⊗ G and G ∧ G are residually p-finite for any prime integer p. For G a limit group or a Coxeter group we show that G ∧ G is residually finite. For G a soluble group of type FP∞ we prove that the groups X(G), ν(G), G ∧ G are virtually residually p-finite for almost all primes p. For G a generalised Baumslag-Solitar group of rank n ≥ 1 that is residually finite, we show that G ∧ G is residually finite. For G = SLn(Z) and G = GLn( ℤ), n ≥ 3, we show that G ∧ G is residually finite.
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BibTeX
@misc{rocha2026residual,
title = {{On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and $SL_n(\Z)$}},
author = {Lucas Barroso Rocha and Dessislava H. Kochloukova},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02243},
eprint = {2610.02243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02243},
}
RIS
TY - UNPB
TI - On the residual finiteness of the non-abelian exterior square of some Artin groups, constructible soluble groups, generalised Baumslag-Solitar groups and SL_n(Z)
AU - Lucas Barroso Rocha
AU - Dessislava H. Kochloukova
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02243
UR - https://arxiv.org/abs/2610.02243
AB - We show that cohomological and homological goodness are equivalent. We prove that if G is a coherent even Artin group or an even Artin group whose underlying graph does not contain 4-clique or a LERF Artin group then G ⊗ G and G ∧ G are residually finite. Furthermore for G a RAAG both G ⊗ G and G ∧ G are residually p-finite for any prime integer p. For G a limit group or a Coxeter group we show that G ∧ G is residually finite. For G a soluble group of type FP_∞ we prove that the groups X(G), ν(G), G ∧ G are virtually residually p-finite for almost all primes p. For G a generalised Baumslag-Solitar group of rank n ≥ 1 that is residually finite, we show that G ∧ G is residually finite. For G = SL_n(Z) and G = GL_n( ℤ), n ≥ 3, we show that G ∧ G is residually finite.
ER -