Diffeomorphism groups of critical regularity

Abstract

Let $M$ be the circle or a compact interval, and let $\alpha=k+\tau\ge1$ 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 $\tau$. If $\alpha\ge1$, we prove that there exists a finitely generated subgroup $G_\alpha\le\mathrm{Diff}_+^\alpha(M)$ with the property that $G_\alpha$ admits no injective homomorphisms into $\mathrm{Diff}_+^\beta(M)$ for all $\beta>\alpha$. If $\alpha>1$, we also show the dual result: there exists a finitely generated group $H_\alpha\le\bigcap_{\beta<\alpha}\mathrm{Diff}_+^\beta(M)$ with the property that $H_\alpha$ admits no injective homomorphisms into $\mathrm{Diff}_+^\alpha(M)$. We can further require that the same properties are inherited by all finite index subgroups, and also by the commutator subgroups, of $G_\alpha$ and $H_\alpha$. The commutator groups of $G_\alpha$ and of $H_\alpha$ are countable simple groups. As a consequence, whenever $1\le\alpha<\beta$ we have a continuum of isomorphism types of finitely generated subgroups of $\mathrm{Diff}_+^{\alpha}(M)$ whose images under arbitrary homomorphisms to $\mathrm{Diff}_+^{\beta}(M)$ 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 $k<\beta$ then there is no nontrivial homomorphism $\mathrm{Diff}_+^k(S^1)\to\mathrm{Diff}_+^{\beta}(S^1)$.