In this thesis, using the calculation of a couple of Kasparov products of asymptotically equivariant cycles, we find the index of an asymptotically equivariant Dirac-Schr¨odinger operator on a hyperbolic manifold. In fact, using the calculation of the Kasparov products of a couple of asymptotically equivariant cycles, we reduce the problem of finding the index to the case in which the manifold is compact and so the problem reduces to the Atiyah-Singer index theorem.

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