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A translation manifold is a manifold whose transition transformations are translations. There is an important connection between the geometry and arithmetic of translation surfaces and dynamics of polygonal billiards. There are also remarkable relations with automorphic forms. In this note we announce results which further develop these connections.doi:10.4310/mrl.1996.v3.n3.a8 fatcat:zmrjhjlnhzaonaclxr2r6r3cjm