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- W2768469774 abstract "Let $M$ be the circle or a compact interval, and let $alpha=k+tauge1$ be a real number such that $k=lfloor alpharfloor$. We write $mathrm{Diff}_+^{alpha}(M)$ for the group of $C^k$ diffeomorphisms of $M$ whose $k^{th}$ derivatives are Holder continuous with exponent $tau$. If $alphage1$, we prove that there exists a finitely generated subgroup $G_alphalemathrm{Diff}_+^alpha(M)$ 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 $H_alphalebigcap_{beta<alpha}mathrm{Diff}_+^beta(M)$ 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 $1lealpha<beta$ 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 $kneq 2$ is an integer and if $k<beta$ then there is no nontrivial homomorphism $mathrm{Diff}_+^k(S^1)tomathrm{Diff}_+^{beta}(S^1)$." @default.
- W2768469774 created "2017-12-04" @default.
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- W2768469774 date "2017-11-15" @default.
- W2768469774 modified "2023-09-27" @default.
- W2768469774 title "Diffeomorphism groups of critical regularity" @default.
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- W2768469774 doi "https://doi.org/10.48550/arxiv.1711.05589" @default.
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