Diffeomorphism groups of critical regularity

Abstract

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