  This paper, together with Part II, expands the results of math.DG/9803051. In
Part I we study the twisted index theory of elliptic operators on orbifold
covering spaces of compact good orbifolds, which are invariant under a
projective action of the orbifold fundamental group. We apply these results to
obtain qualitative results on real and complex hyperbolic spaces in 2 and 4
dimensions, related to generalizations of the Bethe-Sommerfeld conjecture and
the Ten Martini Problem, on the spectrum of self adjoint elliptic operators
which are invariant under a projective action of a discrete cocompact group.
