Diffeomorphism groups of critical regularity

Let $M$ be the circle or a compact interval, and let $α=k+τ\ge1$ be a real number such that $k=\lfloor α\rfloor$. We write $\mathrm{Diff}_+^α(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are Hölder continuous with exponent $τ$. If $α\ge1$, we prove that there exists a finitely generated subgroup $G_α\le\mathrm{Diff}_+^α(M)$ with the property that $G_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^β(M)$ for all $β>α$. If $α>1$, we also show the dual result: there exists a finitely generated group $H_α\le\bigcap_{β<α}\mathrm{Diff}_+^β(M)$ with the property that $H_α$ admits no injective homomorphisms into $\mathrm{Diff}_+^α(M)$. We can further require that the same properties are inherited by all finite index subgroups, and also by the commutator subgroups, of $G_α$ and $H_α$. The commutator groups of $G_α$ and of $H_α$ are countable simple groups. As a consequence, whenever $1\leα<β$ we have a continuum of isomorphism types of finitely generated subgroups of $\mathrm{Diff}_+^α(M)$ whose images under arbitrary homomorphisms to $\mathrm{Diff}_+^β(M)$ are abelian. We give some applications to smoothability of codimension one foliations and to homomorphisms between certain continuous groups of diffeomorphisms. For example, we show that if $k\neq 2$ is an integer and if $k<β$ then there is no nontrivial homomorphism $\mathrm{Diff}_+^k(S^1)\to\mathrm{Diff}_+^β(S^1)$.

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