Mixed 3-manifolds are virtually special

TL;DR

This paper proves that the fundamental groups of certain 3-manifolds, specifically mixed 3-manifolds that are neither graph nor hyperbolic, are virtually special, advancing understanding in 3-manifold topology.

Contribution

It establishes that the fundamental groups of mixed 3-manifolds are virtually special, a significant step in 3-manifold group theory.

Findings

Fundamental groups of mixed 3-manifolds are virtually special.

Extends the class of 3-manifolds known to have virtually special groups.

Provides tools for analyzing 3-manifold groups in geometric topology.

Abstract

Let M be a compact oriented irreducible 3-manifold which is neither a graph manifold nor a hyperbolic manifold. We prove that the fundamental group of M is virtually special.