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Let M be a closed orientable 3-manifold admitting an H2 × R or gSL2(R) geometry, or equivalently a Seifert fibered space with a hyperbolic base 2-orbifold. Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or homotopy equivalent to S1, according as the center of 1(M) is trivial or infinite cyclic. Apart from the remaining case of non-Haken infranilmanifolds, this completes the homeomorphism classifications of Diff(M) and of the space of Seifert fiberings SF(M) for compact orientable aspherical 3-manifolds. We also prove that when M has an H2×R or gSL2(R) geometry and the base orbifold has underlying manifold the 2sphere with three cone points, the inclusion Isom(M) ! Diff(M) is a homotopy equivalence.

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