Thurston geometries from eleven dimensions

All but one of the three-dimensional Thurston geometries can be expressed as N-2 x S-1 or as a U(1) bundle over N-2, where N-2 denotes a two-dimensional Riemannian space of constant curvature. In an M-theoretic framework, these Thurston geometries can be related by Hopf T-duality. The exceptional case is the 'Sol geometry', which results from the dimensional reduction of the decoupling limit of the D3-brane in a background B field.