Small $\text{PSL}(2, \mathbb{F})$ representations of Seifert fiber space groups

TL;DR

The paper establishes bounds on non-abelian PSL(2, F) representations of Seifert fiber space groups and applies this to show lens space recognition is in coNP for such manifolds.

Contribution

It provides explicit bounds on the size of finite fields for non-abelian PSL(2, F) quotients of Seifert fiber space groups and uses this to analyze the complexity of lens space recognition.

Findings

Existence of non-abelian PSL(2, F) quotients with bounded field size

Lens space recognition problem is in coNP for Seifert fiber spaces

Discussion on distinguishing lens spaces from other 3-manifolds

Abstract

Let $M$ be a Seifert fiber space with non-abelian fundamental group and admitting a triangulation with $t$ tetrahedra. We show that there is a non-abelian $\text{PSL}(2, \mathbb{F})$ quotient where $|\mathbb F| < c(2^{20t}3^{120t})$ for an absolute constant $c>0$ and use this to show that the lens space recognition problem lies in coNP for Seifert fiber space input. We end with a discussion of our results in the context of distinguishing lens spaces from other $3$--manifolds more generally.