Group Actions on 1-Manifolds — Question 4

Definitions and mathematical statement

Let $\operatorname{Homeo}_{+}({\mathbb{R}})$ denote the group of orientation-preserving homeomorphisms of the real line. An action of a group $G$ by such homeomorphisms is a homomorphism $\rho\colon G\to\operatorname{Homeo}_{+}({\mathbb{R}})$. Its set of global fixed points is \[ \operatorname{Fix}(\rho)=\{x\in{\mathbb{R}}:\rho(g)(x)=x\text{ for every }g\in G\}. \] Thus the action has no global fixed point precisely when $\operatorname{Fix}(\rho)=\varnothing$; this does not require each nonidentity element to act without fixed points.

The supplied question asks whether there exists an infinite, finitely generated group $G$ that acts on ${\mathbb{R}}$ and whose every orientation-preserving action on ${\mathbb{R}}$ without global fixed points is of type (iii). The universal requirement has the form \[ \forall\rho\in\operatorname{Hom}\left(G,\operatorname{Homeo}_{+}({\mathbb{R}})\right),\qquad \operatorname{Fix}(\rho)=\varnothing \ \Longrightarrow\ \mathsf{Type}_{(iii)}(\rho). \] Here $\mathsf{Type}_{(iii)}$ is an unresolved placeholder for the source's classification: the supplied record does not define type (iii). Consequently this formula specifies the quantifier structure but is not a complete mathematical formulation. The introductory requirement that $G$ "acts on the real line" must also be clarified: the record does not say whether this requires a faithful action, an action without global fixed points, or another nontriviality condition. Merely admitting an arbitrary action would impose no restriction, since every group admits the trivial action. These missing qualifications are retained for review rather than supplied by assumption.

Short English statement

Navas asks whether an infinite, finitely generated group acting on the real line can have all its orientation-preserving actions without global fixed points belong to type (iii). The supplied record omits the definition of that type and the intended strength of the initial action requirement, so a complete precise statement remains unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $800290$, source problem code catalogue entry, title Group Actions on 1-Manifolds --- Question 4. The supplied record attributes the question to Navas (2017) and labels it open; this status is supplied metadata, not an independently verified mathematical assessment.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), Question 4 (#4), https://arxiv.org/abs/1712.06462. The supplied provenance records access on 2026-10-16 and extraction from arXiv LaTeX. These are supplied provenance details; the source has not been independently consulted for this draft.