Diffeomorphism groups of critical regularity. (English) Zbl 1446.57020

Kim, Sang-hyun; Koberda, Thomas

Invent. Math. 221, No. 2, 421-501 (2020).

For a circle or a compact real interval \(M\), let \(\mathrm{Diff}_+^\alpha(M)=\mathrm{Diff}_+^{k + \tau}(M)\) denote the group of orientation preserving \(C^k\) diffeomorphisms of \(M\) whose \(k\)th derivatives are Hölder continuous with exponent \(\tau=\alpha-k\) where \(k\) denotes the integral part of \(\alpha\). As the authors note, the purpose of the present paper is to study the algebraic structure of finitely generated subgroups of \(\mathrm{Diff}_+^\alpha(M)\) for varying \(\alpha\); in particular, the authors give the first construction of finitely generated groups and of countable simple groups in \(\mathrm{Diff}_+^\alpha(M)\) which are not contained in the union \(\bigcup_{\beta > \alpha}\mathrm{Diff}_+^\beta(M)\). More generally, the main result states that there is a continuum of isomorphism types of finitely generated subgroups \(G\) of \(\mathrm{Diff}_+^\alpha(M)\) (with simple commutator subgroups) such that \(G\) admits no injective homomorphisms into \(\bigcup_{\beta > \alpha}\mathrm{Diff}_+^\beta(M)\); dually, they prove that there is a continuum of isomorphism types of finitely generated subgroups \(G\) of \(\bigcap_{\beta < \alpha}\mathrm{Diff}_+^\beta(M)\) such that \(G\) admits no injective homomorphism into \(\mathrm{Diff}_+^\alpha(M)\). Some applications to smoothability of codimension one foliations are given; also, the class of finitely generated subgroups of \(\mathrm{Diff}_+^1(M)\) is not closed under taking finite free products.

Reviewer: Bruno Zimmermann (Trieste)

MSC:

57M60	Group actions on manifolds and cell complexes in low dimensions
20F36	Braid groups; Artin groups
37C05	Dynamical systems involving smooth mappings and diffeomorphisms
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57S05	Topological properties of groups of homeomorphisms or diffeomorphisms

Keywords:

\(C^k\)-diffeomorphism groups of a circle or a closed interval; Hölder continuous derivatives; finitely generated and countable simple subgroups

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