Geometry of cubulated 3-manifolds (Q1910454)

A part of the geometrization conjecture for 3-manifolds reduces to the hyperbolization conjecture that a closed, orientable irreducible 3-manifold \(M\) with infinite fundamental group either is hyperbolic or has a \(\mathbb{Z}^2\) subgroup contained in \(\tau_1(M)\). This paper establishs a weakened form of the hyperbolization conjecture for cubulated 3-manifolds of nonpositive curvature: If \(M\) is a connected, cubulated 3-manifold of nonpositive curvature, then either \(M\) is negatively curved in the sense of Gromov, or \(\pi_1(M)\) has a \(\mathbb{Z}^2\) subgroup. (Note that a cubulation is like a triangulation using cubes and squares instead of tetrahedra and triangles).