Asymptotic expansions for the compact quotients of properly discontinuous group actions

Abstract

Let M be a connected Riemannian manifold and F a group acting isometrically, effectively, and properly discontinuously on M with compact quotient space M F\M. The orbit space M is not necessarily a manifold. Suppose that r" M--> M denotes the associated projection. A function f defined on /(/ is said to be of class CZ(//) if r CZ(M). Since F acts isometrically, the Laplacian A of M is F-invariant and A induces an operator on C(hT/). The Laplacian has a self-ad]oint extension to L(hT/) with pure point spectrum -< he-< '-Our main result, Theorem 4.8, is the asymptotic