Diffeomorphism groups of critical regularity

Let M be a circle or a compact interval, and let α “ k ` τ ě 1 be a real number such that k “ tαu. We write Diff`pMq for the group of orientation preserving Ck diffeomorphisms of M whose kth derivatives are Hölder continuous with exponent τ. We prove that there exists a continuum of isomorphism types of finitely generated subgroups G ď Diff`pMq with the property that G admits no injective homomorphisms into Ť βąα Diff β `pMq. We also show the dual result: there exists a continuum of isomorphism types of finitely generated subgroups G of Ş βăα Diff β `pMq with the property that G admits no injective homomorphisms into Diff`pMq. 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 α ě 1 is a real number not equal to 2, then there is no nontrivial homomorphism Diff`pS 1q Ñ Ť βąα Diff β `pS 1q. Finally, we obtain an independent result that the class of finitely generated subgroups of Diff`pMq is not closed under taking finite free products.