Group Actions with Differentiability Obstructions

Definitions and mathematical statement

For a manifold $M$ and a differentiability index $t$, once the manifold structure, boundary conventions, and meaning of $C^t$ have been specified, let $\operatorname{Diff}^t(M)$ denote the group of bijections $M\to M$ whose maps and inverses have regularity $C^t$. An action of an abstract group $\Gamma$ on $M$ by $C^t$ diffeomorphisms is a homomorphism $\rho:\Gamma\to\operatorname{Diff}^t(M)$. Its kernel is $\ker\rho=\{\gamma\in\Gamma:\rho(\gamma)=\operatorname{id}_M\}$. The action is trivial if $\rho(\Gamma)=\{\operatorname{id}_M\}$, equivalently if $\ker\rho=\Gamma$.

The supplied question asks whether there exists a countable subgroup $\Gamma_{r,M}\leq\operatorname{Diff}^r(M)$, or, more strongly, a finitely generated such subgroup, for which every action on the same manifold $M$ at every higher regularity $s>r$ is trivial, or at least has a "large kernel." The subgroup inclusion supplies a faithful action at regularity $r$.

Conditional on specifying $M$, $r$, and the allowed differentiability indices, the stronger, trivial-action version is \[ \exists\,\Gamma_{r,M}\leq\operatorname{Diff}^r(M),\quad |\Gamma_{r,M}|\leq\aleph_0,\quad \forall s>r\ \forall\rho\in\operatorname{Hom}\!\left(\Gamma_{r,M},\operatorname{Diff}^s(M)\right),\quad \rho(\Gamma_{r,M})=\{\operatorname{id}_M\}. \] The finitely generated version additionally requires a finite set $S\subseteq\Gamma_{r,M}$ with $\Gamma_{r,M}=\langle S\rangle$. These are distinct requested strengths of the existence question, rather than restrictions on the particular faithful action supplied by the inclusion alone.

The source also permits a weaker conclusion that each higher-regularity action has a "large kernel," but supplies no definition of largeness; this alternative cannot be given a precise kernel condition from the record. The range and quantification of $M$ and $r$, including dimension, compactness, boundary conditions, and nonintegral regularity conventions, are unspecified. The record also does not explicitly require $\Gamma_{r,M}\neq\{1\}$; an intended nontriviality condition requires clarification.

Short English statement

The question seeks a countable, preferably finitely generated, group acting faithfully on $M$ with regularity $C^r$, while every action of that group on the same $M$ at any higher regularity is trivial or has a suitably large kernel. The manifold hypotheses, regularity conventions, nontriviality requirement, and meaning of "large" remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005667$, exact source problem code catalogue entry, titled "Group Actions with Differentiability Obstructions," in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Teichmüller Theory: Classical, Higher, Super and Quantum (2024), Oberwolfach Reports, pp. 1843--1892, DOI 10.4171/owr/2023/33. https://doi.org/10.4171/owr/2023/33. The extracted passage identifies p. 1869 and the contribution "First order theory of homeomorphism groups of manifolds, and applications," by Thomas Koberda, joint work with Sang-hyun Kim and J. de la Nuez González. The registry records proposed year 2023 and supplies no separate proposer attribution.
• Kim--Koberda, Structure and regularity of group actions. Supplied literature link: https://arxiv.org/abs/2106.15532.
• Koberda, Diffeomorphism groups of critical regularity. Supplied literature link: https://doi.org/10.1090/noti2252.
• Supplied literature assessment dated 2026-08-22: the record labels the problem "partially solved" and states that the proposed regularity obstruction is proved for compact one-manifolds, including finitely generated examples, while the analogous higher-dimensional existence question remains open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; the precise hypotheses and strength of the cited results require checking.
• The registry labels the statement corrected_verified and reports that the official report was checked directly. That verification claim is supplied provenance and has not been independently verified in this draft.