Graph manifolds with boundary are virtually special

TL;DR

This paper proves that the fundamental groups of certain graph manifolds with boundary are virtually special, meaning they can be embedded into right-angled Artin groups, which implies linearity and other desirable properties.

Contribution

It establishes the virtual specialness of fundamental groups of graph manifolds with boundary by analyzing separability of surface subgroups and double cosets, providing a new class of virtually special groups.

Findings

Fundamental groups of embedded incompressible surfaces are separable.

Double cosets for crossing surfaces are separable.

Fundamental groups of these manifolds are virtually special and linear.

Abstract

Let M be a graph manifold. We prove that fundamental groups of embedded incompressible surfaces in M are separable in the fundamental group of M, and that the double cosets for crossing surfaces are also separable. We deduce that if there is a "sufficient" collection of surfaces in M, then the fundamental group of M is virtually the fundamental group of a special nonpositively curved cube complex. We provide a sufficient collection for graph manifolds with boundary thus proving that their fundamental groups are virtually special, and hence linear.