Group Actions on 1-Manifolds — Question 24

Definitions and mathematical statement

Let $S^1={\mathbb{R}}/{\mathbb{Z}}$, equipped with the standard circle distance \[ d(\bar x,\bar y)=\min_{k\in{\mathbb{Z}}}|x-y-k|. \] A homeomorphism $f:S^1\to S^1$ is Lipschitz with constant at most $L$ if \[ d(f(x),f(y))\leq Ld(x,y)\qquad\text{for all }x,y\in S^1. \] Let $G\leq\operatorname{Diff}^{1}(S^1)$ be a finitely generated group, and suppose that $G$ has no resilient pairs. The supplied record does not define a resilient pair; this hypothesis must therefore retain its source-specific meaning pending verification of the definition.

For a fixed finite generating set $S\subseteq G$, the requested conclusion is \[ \forall\varepsilon>0\ \exists h\in\operatorname{Homeo}(S^1)\ \forall s\in S\ \forall x,y\in S^1,\qquad d\left((h\circ s\circ h^{-1})(x),(h\circ s\circ h^{-1})(y)\right) \leq e^{\varepsilon}d(x,y). \] The question is whether the stated hypotheses imply this conclusion, so that the conjugate group $hGh^{-1}$ consists of Lipschitz homeomorphisms with the displayed bound on its specified generators. The conjugating homeomorphism may depend on $\varepsilon$. The record does not specify the generating set or whether it is symmetric; the displayed bound is imposed on the elements of $S$, with no additional bound on their inverses unless those inverses belong to $S$. No orientation-preserving restriction is stated in the supplied record. A fully self-contained formulation remains contingent on clarification of the resilient-pair hypothesis and the generating-set convention.

Short English statement

Does every finitely generated group of $C^1$ circle diffeomorphisms having no resilient pairs admit, for each $\varepsilon>0$, a topological conjugacy making its generators Lipschitz with constants at most $e^{\varepsilon}$? The source-specific meaning of resilient pairs and the generating-set convention require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800307$, source code catalogue entry, titled Group Actions on 1-Manifolds --- Question 24. The registry attributes the question to Navas, with proposed year 2017, and labels it open. This is a supplied status label, not an independently verified literature assessment.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), https://arxiv.org/abs/1712.06462. The supplied provenance identifies the item as "Question 24 (#23)", records access on 2026-10-16, and gives extraction method arxiv_latex. These provenance details are supplied metadata; the source has not been independently consulted for this draft.