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We prove the automorphism groups of the compact measure hyperbolic manifolds are discrete. We also obtain the following nontrivial fact as a consequence of Theorems 1 and 2. For some related results one can consult a recent survey article of Kobayashi (Intrinsic distance, measure and geometric function theory, Bull.doi:10.1090/s0002-9939-1977-0425194-0 fatcat:yuqkr4dlljeyhof564let6jske