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arXiv · 2611.00142

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 $\mathrm{PGL}_2(R),$ 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 $\mathrm{SL}_2, \mathrm{GL}_2, \mathrm{PSL}_2$, or $\mathrm{PGL}_2$ 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 $\mathrm{SL}_2$ and $\mathrm{GL}_2$. Here, we provide a modified argument for the $\mathrm{PSL}_2$ and $\mathrm{PGL}_2$ cases.

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Montek Singh Gill. 2026-11-03. Erratum to "Linear representations of 3-manifold groups over rings". https://arxiv.org/abs/2611.00142

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