Virtual cubulation of nonpositively curved graph manifolds

TL;DR

This paper proves that nonpositively curved graph manifolds are characterized by their fundamental groups virtually embedding into right-angled Artin groups, leading to linearity of their fundamental groups.

Contribution

It establishes a precise criterion linking nonpositive curvature of graph manifolds to embeddings into right-angled Artin groups, a novel characterization in geometric group theory.

Findings

Nonpositively curved graph manifolds have fundamental groups that virtually embed into right-angled Artin groups.

Such manifolds have linear fundamental groups.

The characterization provides a new understanding of the geometry and algebra of these manifolds.

Abstract

In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.