Finite quotients of 3-manifold groups

TL;DR

This paper investigates which finite groups can appear as quotients of 3-manifold groups, providing criteria based on group cohomology and probabilistic models, and introduces new methods for understanding the distribution of such quotients.

Contribution

It characterizes the existence of certain finite quotients of 3-manifold groups using group cohomology and probabilistic techniques, including a novel distribution formula for random 3-manifold groups.

Findings

Non-existence of certain quotients proven via topological generalizations.

Existence of specific quotients demonstrated using probabilistic methods.

Introduces a new distribution formula for random 3-manifold groups.

Abstract

For $G$ and $H_1,\dots, H_n$ finite groups, does there exist a $3$-manifold group with $G$ as a quotient but no $H_i$ as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with topological results generalizing the theory of semicharacteristics. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the (profinite completion of) the fundamental group of a random 3-manifold in the Dunfield-Thurston model of random Heegaard splittings as the genus goes to infinity. We believe this is the first construction of a new distribution of random groups from its moments.