Let $M$ be the circle or a compact interval, and let $\alpha=k+\tau\ge2$ be a real number such that $k=\lfloor \alpha\rfloor$. We write $\mathrm{Diff}_+^{\alpha}(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are H\"older continuous with exponent $\tau$. We prove that there exists a finitely generated subgroup $G_\alpha$ of $\mathrm{Diff}_+^{\alpha}(M)$ with the property that $G_\alpha$ admits no injective homomorphisms into $\mathrm{Diff}_+^{\beta}(M)$ whenever $\beta>\alpha$. We can further require the same property for all finite index subgroups of $G_\alpha$, and also for $[G_\alpha,G_\alpha]$, the latter of which is a countable simple group. This implies the existence of a continuum of isomorphism types of finitely generated subgroups (and countable simple subgroups) of $\mathrm{Diff}_+^{\alpha}(M)$ which admit no injective homomorphisms into $\mathrm{Diff}_+^{\beta}(M)$. We give some applications to smoothability of codimension one foliations and 查看全文>>