Group Actions on 1-Manifolds — Question 8

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 every $a,g,h\in G$, \[ g<h\implies ag<ah. \] The condition that there is no nontrivial homomorphism into ${\mathbb{Z}}$ means that every group homomorphism $\varphi:G\to({\mathbb{Z}},+)$ satisfies $\varphi(g)=0$ for every $g\in G$.

Interpret $\mathrm H_b^2(G,{\mathbb{R}})$ as ordinary bounded group cohomology with trivial real coefficients. Explicitly, let \[ C_b^n(G,{\mathbb{R}})=\left\{f:G^n\to{\mathbb{R}}:\sup_{\mathbf g\in G^n}|f(\mathbf g)|<\infty\right\}. \] For $f\in C_b^1(G,{\mathbb{R}})$ and $c\in C_b^2(G,{\mathbb{R}})$, set \[ (\delta_1f)(g,h)=f(h)-f(gh)+f(g), \] \[ (\delta_2c)(g,h,k)=c(h,k)-c(gh,k)+c(g,hk)-c(g,h). \] Then \[ \mathrm H_b^2(G,{\mathbb{R}})=\ker\delta_2/\operatorname{im}\delta_1. \] Thus its vanishing means that every bounded real-valued $2$-cocycle is the coboundary of a bounded real-valued $1$-cochain.

The supplied question, attributed to Navas (2017), asks whether there exists a finitely generated, left-orderable group $G$ such that \[ \operatorname{Hom}(G,{\mathbb{Z}})=\{0\}, \qquad \mathrm H_b^2(G,{\mathbb{R}})=0. \] The supplied formulation does not require $G$ to be nontrivial or infinite. No such hypothesis has been added here; whether an exclusion of the trivial group was intended remains unresolved and requires checking against the source.

Short English statement

Navas asks for a finitely generated group admitting a left-invariant total order, having no nonzero homomorphism to ${\mathbb{Z}}$, and having vanishing second bounded cohomology with trivial real coefficients. The supplied statement leaves unresolved whether the group must be nontrivial.

Sources

• Local UnsolvedMath registry, permanent ID $800294$, exact source problem code catalogue entry, title Group Actions on 1-Manifolds --- Question 8; supplied under AMR Open Problem Lists. The record attributes the question to Navas, with proposed year 2017.
• Navas, Group Actions on 1-Manifolds: A List of Very Concrete Open Questions (2017), Question 8 (#8). Supplied source URL: https://arxiv.org/abs/1712.06462. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not independently verified access or source inspection.
• The local record labels the problem open, with update timestamp 2026-10-16T14:00:00Z and upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. This status is supplied curation, not an independently verified mathematical or literature assessment.