Diffeomorphism groups of critical regularity

Abstract

Let M be a circle or a compact interval, and let alpha =k+tau ge 1 be a real number such that k=lfloor alpha rfloor . We write {{,mathrm{Diff},}}_+^{alpha }(M) for the group of orientation preserving C^k diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent tau . We prove that there exists a continuum of isomorphism types of finitely generated subgroups Gle {{,mathrm{Diff},}}_+^alpha (M) with the property that G admits no injective homomorphisms into bigcup _{beta >alpha }{{,mathrm{Diff},}}_+^beta (M). We also show the dual result: there exists a continuum of isomorphism types of finitely generated subgroups G of bigcap _{beta <alpha }{{,mathrm{Diff},}}_+^beta (M) with the property that G admits no injective homomorphisms into {{,mathrm{Diff},}}_+^alpha (M). The groups G are constructed so that their commutator groups are simple. 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 alpha ge 1 is a real number not equal to 2, then there is no nontrivial homomorphism {{,mathrm{Diff},}}_+^alpha (S^1)rightarrow bigcup _{beta >alpha }{{,mathrm{Diff},}}_+^{beta }(S^1). Finally, we obtain an independent result that the class of finitely generated subgroups of {{,mathrm{Diff},}}_+^1(M) is not closed under taking finite free products.