2-Linearizability of Geometric 3-Manifold Groups Over Commutative Rings
Abstract
The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form PGL(2,R), where R is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but 𝕊3 and SL2̃. In the case of 𝕊3, the conjecture holds if we replace PGL(2,R) with GL(2,R). A spherical counterexample for the projective variant is the Poincaré homology sphere Σ(2,3,5). In the case of SL2̃, the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere Σ(2,3,7).
Related
- Automorphisms of Twisted Chevalley Groups over Commutative Rings
- Erratum to "Linear representations of 3-manifold groups over rings"
- On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings
- Commutative coherent rings with 𝒢𝒫not⊆𝒢ℱ
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BibTeX
@misc{gill20262linearizability,
title = {{2-Linearizability of Geometric 3-Manifold Groups Over Commutative Rings}},
author = {Montek Singh Gill},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.01532},
eprint = {2608.01532},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01532},
}
RIS
TY - UNPB
TI - 2-Linearizability of Geometric 3-Manifold Groups Over Commutative Rings
AU - Montek Singh Gill
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2608.01532
UR - https://arxiv.org/abs/2608.01532
AB - The fundamental groups of compact 3-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by only allowing finite groups of the form PGL(2,R), where R is a finite commutative ring. In earlier work, this conjecture was disproven in full generality. The conjecture arose in the context of orientable connected compact 3-manifolds which are geometrizable. By constructing explicit faithful linear representations using rings with nilpotent elements, we demonstrate that the conjecture holds for six of the eight Thurston model geometries, namely all but 𝕊^3 and SL_2̃. In the case of 𝕊^3, the conjecture holds if we replace PGL(2,R) with GL(2,R). A spherical counterexample for the projective variant is the Poincaré homology sphere Σ(2,3,5). In the case of SL_2̃, the conjecture fails to hold for both the projective and non-projective variants; a counterxample is provided by the Brieskorn sphere Σ(2,3,7).
ER -