Translation representations for automorphic solutions of the wave equation in non‐euclidean spaces. III

In Part III we show that the translation representations defined in Part I, and shown to be complete in Part II, are incoming and outgoing in the sense of propagation of signals along rays. We give a new proof of the absolute continuity of the spectrum below ‐¼ n 2 , and point out its implications for local energy decay. We give a new proof of completeness in dimensions 2 and 3 when energy is positive. Finally, we define and prove completeness of the translation representations when the metric is perturbed on a compact set.