Diffeomorphism groups of critical regularity

• 成果类型: Article
• 署名作者: Kim, Sang-hyun; Koberda, Thomas
• 署名单位: Korea Institute for Advanced Study (KIAS); University of Virginia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00953-y
• 发表日期: 2020
• 页码: 421-501
• 关键词: locally indicable group; c-1 diffeomorphisms; sharp regularity; nilpotent group; homeomorphisms; HYPERBOLICITY; smoothability; homomorphisms; Commutators; continuity

摘要

Let M be a circle or a compact interval, and let alpha = k + tau >= 1 be a real number such that k = [alpha]. We write Diff(+)(alpha)(M) for the group of orientation preserving C-k diffeomorphisms of M whose kth derivatives are Holder continuous with exponent tau. We prove that there exists a continuum of isomorphism types of finitely generated subgroups G <= Diff(+)(alpha)(M) with the property that G admits no injective homomorphisms into boolean OR(beta>alpha) Diff(+)(beta)(M). We also show the dual result: there exists a continuum of isomorphism types of finitely generated subgroups G of boolean AND(beta>alpha) Diff(+)(beta)(M) with the property that G admits no injective homomorphisms into 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 >= 1 is a real number not equal to 2, then there is no nontrivial homomorphism Diff(+)(alpha)(S-1) -> boolean OR(beta>alpha) Diff(+)(beta)(S-1). Finally, we obtain an independent result that the class of finitely generated subgroups of Diff(+)(1)( M) is not closed under taking finite free products.