  In this paper, we study both the continuous model and the discrete model of
the Quantum Hall Effect (QHE) on the hyperbolic plane. The Hall conductivity is
identified as a geometric invariant associated to an imprimitivity algebra of
observables. We define a twisted analogue of the Kasparov map, which enables us
to use the pairing between $K$-theory and cyclic cohomology theory, to identify
this geometric invariant with a topological index, thereby proving the
integrality of the Hall conductivity in this case.
