Group Actions on 1-Manifolds — Question 20

Definitions and mathematical statement

Let $S^1={\mathbb{R}}/{\mathbb{Z}}$, and let $\operatorname{Diff}^{\infty}_{+}(S^1)$ be the group of smooth orientation-preserving diffeomorphisms of the circle. An action of ${\mathbb{Z}}^2$ is a homomorphism \[ \rho:{\mathbb{Z}}^2\longrightarrow\operatorname{Diff}^{\infty}_{+}(S^1). \] Writing $f=\rho(1,0)$ and $g=\rho(0,1)$, this means $fg=gf$. The action is faithful if \[ f^m g^n=\operatorname{id}_{S^1}\quad\Longrightarrow\quad(m,n)=(0,0) \qquad\text{for all }(m,n)\in{\mathbb{Z}}^2. \] The supplied question, attributed to Navas (2017), asks whether there exists such a faithful action that is $C^{\infty}$ structurally stable.

The supplied record does not define structural stability. One possible precise interpretation uses the space \[ \mathcal A=\{(f,g)\in\operatorname{Diff}^{\infty}_{+}(S^1)^2:fg=gf\} \] with the subspace topology induced by the product $C^{\infty}$ topology. Under the interpretation that stability means topological conjugacy among all nearby actions with the generators fixed, the question becomes: does there exist a faithful $(f,g)\in\mathcal A$ and a neighborhood $U$ of $(f,g)$ in $\mathcal A$ such that, for every $(\widetilde f,\widetilde g)\in U$, there exists a homeomorphism $h:S^1\to S^1$ satisfying \[ h\circ f=\widetilde f\circ h, \qquad h\circ g=\widetilde g\circ h? \] This interpretation requires source confirmation: the record does not specify the regularity or orientation of the conjugacy, whether perturbations are restricted to faithful actions, or whether the conjugacy must be close to the identity. These unresolved choices are part of the formulation, and the displayed interpretation is not asserted to be the source's intended definition.

Short English statement

Navas asks whether two commuting smooth orientation-preserving circle diffeomorphisms can generate a faithful ${\mathbb{Z}}^2$-action that is $C^{\infty}$ structurally stable. The supplied record leaves the precise meaning of structural stability unresolved. Its registry status is supplied as open.

Sources

• Local UnsolvedMath registry, permanent ID $800303$, source code catalogue entry, title Group Actions on 1-Manifolds --- Question 20; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied record attributes the problem to Navas, with proposed year 2017, and labels it open; that status is supplied curation, not an independently verified mathematical assessment.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), source item recorded as "Question 20 (#19)": https://arxiv.org/abs/1712.06462. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not a claim of independent access or verification.