2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79406This 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.34 pages, LaTexDifferential GeometryMathematical PhysicsK-Theory and HomologyOperator Algebras46L85, 81Q10text