Degenerate spacetimes in first order gravity

We present a systematic framework to obtain the most general solutions of the equations of motion in first order gravity theory with degenerate tetrads. There are many possible solutions. Generically, these exhibit non-vanishing torsion even in the absence of any matter coupling. These solutions are shown to contain a special set of eight configurations which are associated with the homogeneous model three-geometries of Thurston.