# Linear representations of 3-manifold groups over rings

## TL;DR

This paper explores the representation of 3-manifold groups over finite rings, examining Luo's conjecture that these groups can be faithfully represented into finite groups like PGL(2,R), and provides examples and a counterexample.

## Contribution

It offers an equivalent formulation of Luo's conjecture and presents new examples and a counterexample related to representations over finite rings.

## Key findings

- Counterexample disproves the conjecture in some cases
- Examples support the conjecture in specific instances
- Equivalent reformulation of Luo's conjecture provided

## 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 some finite commutative ring with identity. We give an equivalent formulation of Luo's conjecture via faithful representations and provide various examples and a counterexample.