Negative curvature for geometric complexes

Let $X$ be a finite, irreducible, geometric, branched 2-complex. Write $\partial M(X)$ for the boundary of its associated 3-manifold and $\pi_1(X)$ for its fundamental group.

Negative curvature for geometric complexes. If $\sigma_+(X)=0$ then either $\partial M(X)$ has a toroidal boundary component or $\pi_1(X)$ has a $\mathbb{Z}^2$ subgroup.

The source describes this as a weak form of hyperbolisation for Haken 3-manifolds; no resolution is supplied.