Cocompact cubulations of mixed 3-manifolds

TL;DR

This paper classifies mixed 3-manifolds with virtually compact special fundamental groups, showing they are chargeless, which depends on the triviality of Euler numbers in Seifert fibered blocks, completing the existing classification.

Contribution

It extends the classification of 3-manifolds with virtually compact special fundamental groups to include mixed manifolds, identifying chargelessness as the key condition.

Findings

Mixed 3-manifolds are virtually compact special iff they are chargeless.

Chargelessness depends on trivial Euler numbers in Seifert fibered blocks.

Complete classification of such manifolds in the mixed case.

Abstract

In this paper, we complete the classification of which compact 3-manifolds have a virtually compact special fundamental group by addressing the case of mixed 3-manifolds. A compact aspherical 3-manifold is mixed if has at least one JSJ torus and at least one hyperbolic block. We show the fundamental group of a mixed manifold M is virtually compact special iff M is chargeless, i.e. each interior Seifert fibered block has a trivial Euler number relative to the fibers of adjacent blocks.