Catalog · 10.5281/zenodo.23731787
A finitely presented nonindicable subgroup of a free profinite group
- Martino Lupini · University of Bologna
28 October 2026 · CC BY 4.0 · latest of 2 versions
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A finitely presented nonindicable subgroup of a free profinite group
Martino Lupini
Abstract
We construct an infinite, two-generated, finitely presented subgroup H of the free profinite group of rank two whose abelianization is (ℤ/60ℤ)^2. Thus free profinite groups of rank at least two fail to be locally indicable, disproving a conjecture of Jaikin-Zapirain. The group H has a normal subgroup of index 60 isomorphic to F_29 × F_19 × F_11, where F_r denotes the abstract free group of rank r. More generally, every nontrivial finite perfect group is a quotient of a finitely presented torsion-free group with finite abelianization that embeds in the free profinite group of rank two. The construction combines free products of finite cyclic groups over a common finite quotient, completing a different free kernel at each prime.
GitHub repository: https://github.com/eulogica/free-profinite-local-indicability
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a5_quotient_pairs.py
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check_abelianization.py
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- 10.5281/zenodo.23731787
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BibTeX
@unpublished{Lupini202623731787,
title = {A finitely presented nonindicable subgroup of a free profinite group},
author = {Martino Lupini},
year = {2026},
doi = {10.5281/zenodo.23731787},
note = {Zenodo deposit, latest of 2}
}