Elementary equivalence and diffeomorphism groups of smooth manifolds

Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$–diffeomorphism group of $M$ is elementarily equivalent to the $C^s$–diffeomorphism group of $N$ for some $r,s\in [1,\mathrm{\infty })\cup \{0,\mathrm{\infty }\}$, then $r=s$ and $M$ and $N$ are $C^r$–diffeomorphic. This strengthens a previously known result by Takens and Filipkiewicz, which asserts that for integer regularities, a group isomorphism between diffeomorphism groups of closed manifolds necessarily arises from a diffeomorphism of the underlying manifolds. We prove an analogous result for groups of diffeomorphisms preserving smooth volume forms, in dimension at least two.