We develop a theory of pseudodifferential operators and index theory on open manifolds, initiated by J. Roe, G. Yi, and A. Kordyukov. We generalize Guoliang Yu's index theorem for graded generalized Dirac operators on uniformly contractible manifolds with subexponential growth to elliptic uniform pseudodifferential operators on open manifolds with isometric and cocompact group action. The indices of elliptic uniform pseudodifferential operators will live in the K-K−theory of Uniform Roe algebra.