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We prove that in dimensions not equal to 4, 5, or 7, the homology andhomotopy groups of the classifying space of the topological group ofdiffeomorphisms of a disk fixing the boundary are finitely generated in eachdegree. The proof uses homological stability, embedding calculus and thearithmeticity of mapping class groups. From this we deduce similar results forthe homeomorphisms of $R^n$ and various types of automorphisms of 2-connectedmanifolds.

Some finiteness results for groups of automorphisms of manifolds