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Thurston approaches 3-manifolds by ``cutting them up'' (the Haken decomposition along incompressible surfaces) into pieces which each admit a locally homogeneous geometric structure, modelled on a homogeneous space with an invariant Riemannian metric. A compact, simply connected manifold with such a structure is homogeneous. But a Lie group is homogeneous under its own left translations (or right translations), with many invariant Riemannian metrics (given by left, or right, translation of any positive definite inner product on any one tangent space. So Thurston's programme contains your programme as a special case.