Group Actions on 1-Manifolds — Question 7

Definitions and mathematical statement

A group $G$ is finitely generated if $G=\langle S\rangle$ for some finite subset $S\subseteq G$. It is left-orderable if it admits a strict total order $<$ such that, for all $a,b,g\in G$, $a<b$ implies $ga<gb$. Let $F_2$ denote the free group on two generators, and let $\operatorname{Hom}(G,{\mathbb{Z}})$ denote the set of group homomorphisms from $G$ to the additive group of integers. Here $0$ denotes the homomorphism sending every element to zero.

Under the interpretation that "without free subgroups" means without nonabelian free subgroups, the supplied question asks whether there exists a group $G$ such that \[ \begin{gathered} G\text{ is finitely generated and left-orderable},\\ \nexists\,H\leq G\text{ with }H\cong F_2,\\ \operatorname{Hom}(G,{\mathbb{Z}})=\{0\}. \end{gathered} \] Excluding $F_2$ excludes every nonabelian free subgroup, since every nonabelian free group contains a subgroup isomorphic to $F_2$.

Unresolved qualifications. The supplied wording does not specify whether "free subgroups" means nonabelian free subgroups or also includes infinite cyclic subgroups. Under the latter reading, the subgroup condition would instead exclude every subgroup isomorphic to a free group of positive rank, including ${\mathbb{Z}}$. Furthermore, the supplied statement does not require $G$ to be nontrivial or infinite. Neither requirement has been added here; the intended target requires clarification.

Short English statement

The question attributed to Navas asks for a finitely generated left-orderable group with no free subgroups and no nonzero homomorphism to ${\mathbb{Z}}$. The meaning of "free subgroups" and the absence of a nontriviality requirement remain unresolved in the supplied formulation.

Sources

• Local UnsolvedMath registry, permanent ID $800293$, source code catalogue entry, title "Group Actions on 1-Manifolds --- Question 7." The record attributes the question to Navas (2017) and labels it open; this is a supplied status label, not an independently verified mathematical assessment.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), Question 7 (#7). Supplied source URL: https://arxiv.org/abs/1712.06462. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; neither source access nor extraction fidelity has been independently verified for this draft.