Foliations and Diffeomorphism Groups — Question 45

Definitions and mathematical statement

Let $G$ be a finitely generated amenable group that is orderable in the sense intended by the source. Here amenability means that there exists a finitely additive probability measure $m\colon\mathcal P(G)\to[0,1]$ on all subsets of $G$, invariant under left translation. The supplied record does not specify whether "orderable" means left-orderable or bi-orderable; these hypotheses are therefore not identified with one another here.

Let $\operatorname{Homeo}_+({\mathbb{R}})$ denote the group of orientation-preserving homeomorphisms of ${\mathbb{R}}$, and let \[ \varphi\colon G\longrightarrow\operatorname{Homeo}_+({\mathbb{R}}) \] be a group homomorphism. Assume that the action has no global fixed point, meaning \[ \bigcap_{g\in G}\{x\in{\mathbb{R}}:\varphi(g)(x)=x\}=\varnothing, \] or, equivalently, $\forall x\in{\mathbb{R}}\;\exists g\in G$ such that $\varphi(g)(x)\ne x$. No faithfulness assumption is supplied.

An orientation-preserving affine action is a homomorphism $\rho\colon G\to\operatorname{Aff}_+({\mathbb{R}})$, where \[ \operatorname{Aff}_+({\mathbb{R}})=\{x\mapsto ax+b:a>0,\ b\in{\mathbb{R}}\}. \] Thus $\rho(g)(x)=a_gx+b_g$, with $a_g>0$, $a_{gh}=a_ga_h$, and $b_{gh}=a_gb_h+b_g$.

The source asks whether, for every such $G$ and $\varphi$, the following dichotomy holds: \[ \left[\varphi\text{ is semi-conjugate to an affine action}\right] \quad\text{or}\quad \left[\varphi\text{ is laminar}\right]. \] The disjunction is not asserted to be exclusive. The terms "semi-conjugate" and "laminar" are retained from the source. The supplied record does not define laminar actions or specify the direction and regularity, monotonicity, surjectivity, or properness requirements of a semi-conjugacy. Consequently this is a statement-only transcription of the proposed dichotomy; its fully precise formulation requires those source conventions, as well as the intended meaning of orderability.

Short English statement

For every finitely generated amenable orderable group acting on ${\mathbb{R}}$ by orientation-preserving homeomorphisms with no common fixed point, must the action be semi-conjugate to an affine action or be laminar? The supplied record leaves the meanings of orderability, semi-conjugacy, and laminarity insufficiently specified for a complete formal statement.

Sources

• UnsolvedMath local registry, permanent ID $800104$, exact source problem code catalogue entry, titled "Foliations and Diffeomorphism Groups --- Question 45." The supplied registry attributes the problem to Nariman and Yazdi, proposed in 2025, and labels it open; this label is supplied curation, not an independently verified mathematical or literature assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 45. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance details are supplied and were not independently verified.