All Dehn Fillings of the Whitehead Link Complement are Tetrahedron Manifolds

In this paper we show that Dehn surgeries on the oriented components of the Whitehead link yield tetrahedron manifolds of Heegaard genus $\le 2$. As a consequence, the eight homogeneous Thurston 3-geometries are realized by tetrahedron manifolds of Heegaard genus $\le 2$. The proof is based on techniques of Combinatorial Group Theory, and geometric presentations of manifold fundamental groups.