PreprintCC BY 4.0arXiv
Finite presentations of metabelian groups: effective enumeration via Laurent relations
Abstract
For an ordinary finite presentation P=⟨ x1,…,xn| R⟩, put G(P)=Fn/⟨⟨ R⟩⟩. We construct a primitive-recursive predicate V with G(P)''=1 Longleftrightarrow ∃ c∈ℕ: V(P,c)=1. Thus finite presentations of metabelian groups are recursively enumerable, answering Kourovka Problem 17.124. An effective form of the Bieri-Strebel covering construction, using signed Laurent relations and rational separation, gives a family of finitely presented metabelian groups cofinal under epimorphisms. Products of conjugates of defining relators witness these epimorphisms. The construction and enumeration theorem are formalized in Lean 4.
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BibTeX
@misc{jayadevan2026finite,
title = {{Finite presentations of metabelian groups: effective enumeration via Laurent relations}},
author = {Achyuth Jayadevan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10281},
eprint = {2609.10281},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10281},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite presentations of metabelian groups: effective enumeration via Laurent relations
AU - Achyuth Jayadevan
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10281
UR - https://arxiv.org/abs/2609.10281
AB - For an ordinary finite presentation P=⟨ x_1,…,x_n| R⟩, put G(P)=F_n/⟨⟨ R⟩⟩. We construct a primitive-recursive predicate V with G(P)''=1 Longleftrightarrow ∃ c∈ℕ: V(P,c)=1. Thus finite presentations of metabelian groups are recursively enumerable, answering Kourovka Problem 17.124. An effective form of the Bieri-Strebel covering construction, using signed Laurent relations and rational separation, gives a family of finitely presented metabelian groups cofinal under epimorphisms. Products of conjugates of defining relators witness these epimorphisms. The construction and enumeration theorem are formalized in Lean 4.
ER -