Undecidability of the elementary theory of groups of measure-preserving transformations (Q1280689)

Let \(G\) be the group of invertible transformations of \((0,1)\) preserving the Lebesgue measure. Two transformations are equal if they differ only on a set of measure zero. Let \(L\) be the language of group theory containing variables \(f,g,\ldots\) for elements of the group, the group product \(fg\) (composition), the equality \(f=g\), connectives and quantifiers. Theorem: There is a translation (effective procedure) that for any closed arithmetic formula produces a closed formula in \(L\) which is true if and only if the arithmetic formula is. Corollary: The elementary theory of groups of measure-preserving transformations is nonsolvable.