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Erratum to "Linear representations of 3-manifold groups over rings"

  • Montek Singh Gill
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Abstract

The fundamental groups of compact $3$-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by allowing only finite groups of the form PGL2(R), where R is a finite commutative ring with identity. In a prior paper, "Linear representations of $3$-manifold groups over rings", we, together with two co-authors, gave an equivalent formulation of Luo's conjecture via faithful representations, investigated variants of the conjecture where the target group is of type SL2, GL2, PSL2, or PGL2 as in the original conjecture, and claimed to provide a $3$-manifold group which violated all four variants of the conjecture. We have discovered a flaw in the work, such that the existing proof is valid only for SL2 and GL2. Here, we provide a modified argument for the PSL2 and PGL2 cases.

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Cite

BibTeXRIS
BibTeX
@misc{gill2026erratum,
  title = {{Erratum to "Linear representations of 3-manifold groups over rings"}},
  author = {Montek Singh Gill},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00142},
  eprint = {2611.00142},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00142},
}
RIS
TY  - UNPB
TI  - Erratum to "Linear representations of 3-manifold groups over rings"
AU  - Montek Singh Gill
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00142
UR  - https://arxiv.org/abs/2611.00142
AB  - The fundamental groups of compact $3$-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by allowing only finite groups of the form PGL_2(R), where R is a finite commutative ring with identity. In a prior paper, "Linear representations of $3$-manifold groups over rings", we, together with two co-authors, gave an equivalent formulation of Luo's conjecture via faithful representations, investigated variants of the conjecture where the target group is of type SL_2, GL_2, PSL_2, or PGL_2 as in the original conjecture, and claimed to provide a $3$-manifold group which violated all four variants of the conjecture. We have discovered a flaw in the work, such that the existing proof is valid only for SL_2 and GL_2. Here, we provide a modified argument for the PSL_2 and PGL_2 cases.
ER  -