Let 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 k th 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.

中文翻译：

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临界正则微分同胚群

令 M 为圆或紧区间，令 $$\alpha =k+\tau \ge 1$$ α = k + τ ≥ 1 为实数，使得 $$k=\lfloor \alpha \rfloor $$ k = ⌊ α ⌋ 。我们写 $${{\,\mathrm{Diff}\,}}_+^{\alpha }(M)$$ Diff + α ( M ) 为一组方向保持 $$C^k$$ C k M 的微分同胚，其 k 次导数是 Hölder 连续指数 $$\tau $$ τ 。我们证明存在有限生成子群的同构类型的连续统 $$G\le {{\,\mathrm{Diff}\,}}_+^\alpha (M)$$ G ≤ Diff + α ( M )具有 G 不允许单射同态进入 $$\bigcup _{\beta >\alpha }{{\,\mathrm{Diff}\,}}_+^\beta (M)$$ ⋃ β > α Diff + β (M)。我们还展示了对偶结果：存在 $$\bigcap _{\beta <\alpha }{{\,\mathrm{Diff}\, 的有限生成子群 G 的同构类型的连续统，}}_+^\beta (M)$$ ⋂ β < α Diff + β ( M ) 具有 G 不允许注入同态到 $${{\,\mathrm{Diff}\,}}_+^ 的性质\alpha (M)$$ Diff + α ( M ) 。构造群 G 使得它们的交换子群是简单的。我们对codimension one foliations 的平滑性和某些连续微分同胚群之间的同态给出了一些应用。例如，我们证明如果 $$\alpha \ge 1$$ α ≥ 1 是一个不等于 2 的实数，那么不存在非平凡同态 $${{\,\mathrm{Diff}\,}}_ +^\alpha (S^1)\rightarrow \bigcup _{\beta >\alpha }{{\,\mathrm{Diff}\,}}_+^{\beta }(S^1)$$ Diff + α (S 1) → ⋃ β > α Diff + β (S 1)。最后，我们得到一个独立的结果，$${{\,\mathrm{Diff}\ 的有限生成子群的类，