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  1. For every compact, connected manifold M M , we prove the existence of a sentence ϕ<#comment/> M \phi _M in the language of groups such that the homeomorphism group of another compact manifold N N satisfies ϕ<#comment/> M \phi _M if and only if N N is homeomorphic to M M . We prove an analogous statement for groups of homeomorphisms preserving Oxtoby–Ulam probability measures. 
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