Uniform first order interpretation of the second order theory of countable groups of homeomorphisms

Abstract: We show that the first order theory of the homeomorphism group of a compact manifold interprets the full second order theory of countable groups of homeomorphisms of the manifold. The interpretation is uniform across manifolds of bounded dimension. As a consequence, many classical problems in group theory and geometry (e.g.~the linearity of mapping classes of compact $2$--manifolds) are encoded as elementary properties homeomorphism groups of manifolds. Furthermore, the homeomorphism group interprets the Borel and projective hierarchies of the homeomorphism group. Finally, we show that the collection of sentences that isolate the homeomorphism group of a particular manifold, or that isolate the homeomorphism groups of manifolds in general, is not definable in arithmetic, and that membership of particular sentences in these collections cannot be proved in ZFC.