A 3-manifold group which is not four dimensional linear

TL;DR

This paper constructs examples of 3-manifold groups that cannot be embedded into GL(4,k) for any field k, addressing a long-standing question in 3-manifold topology and linear group theory.

Contribution

It provides explicit examples of 3-manifold groups that are not four-dimensional linear, resolving a question from the Kirby problem list.

Findings

Existence of 3-manifold groups not embeddable in GL(4,k)

Answers a question posed by William Thurston in 1977

Advances understanding of the linearity properties of 3-manifold groups

Abstract

We give examples of closed orientable graph 3-manifolds with fundamental group which is not a subgroup of GL(4,k) for any field k. This answers a question in the Kirby problem list from 1977 which is credited to the late William Thurston.