Group Actions on 1-Manifolds — Question 6

Definitions and mathematical statement

Let $G$ be a group. A left order on $G$ is a strict total order $\prec$ satisfying \[ a\prec b\ \Longrightarrow\ ca\prec cb \qquad(a,b,c\in G). \] Write $\operatorname{LO}(G)$ for the set of all left orders on $G$, and call $G$ left-orderable when this set is nonempty. Give $\operatorname{LO}(G)$ the topology generated by the sets \[ U_{a,b}=\{\prec\in\operatorname{LO}(G):a\prec b\}, \qquad a,b\in G; \] thus a basic neighborhood specifies finitely many order comparisons. The conjugacy action of $G$ on $\operatorname{LO}(G)$ is defined by \[ a\mathrel{(g\cdot\prec)}b \quad\Longleftrightarrow\quad g^{-1}ag\prec g^{-1}bg \qquad(a,b,g\in G). \] This action is minimal if every orbit is dense, meaning \[ \overline{\{g\cdot\prec:g\in G\}}=\operatorname{LO}(G) \qquad\text{for every }\prec\in\operatorname{LO}(G). \] The question attributed to Navas asks whether there exists a group $G$, generated by a finite subset and left-orderable, whose conjugacy action on $\operatorname{LO}(G)$ is minimal.

Unresolved qualification. The supplied statement does not require $G$ to be nontrivial or infinite, and imposes no cardinality condition on $\operatorname{LO}(G)$. These restrictions have not been added. Whether an additional restriction was intended requires source clarification.

Short English statement

Does there exist a finitely generated left-orderable group such that conjugating any one of its left orders produces a dense orbit in the space of all its left orders? The supplied wording leaves possible restrictions excluding trivial cases unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $800292$, source problem code catalogue entry, title Group Actions on 1-Manifolds --- Question 6; collection: AMR Open Problem Lists. The supplied record attributes the question to Navas, with proposed year 2017, and labels its status as 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), Question 6 (#6). Supplied source URL: https://arxiv.org/abs/1712.06462. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are source-record provenance claims, not a claim of independent access or verification for this draft.