A Nonamenable Extension Beyond Thompson's Group

Definitions and mathematical statement

Let $\operatorname{Homeo}^{+}({\mathbb{R}})$ be the group of increasing homeomorphisms of ${\mathbb{R}}$, with composition as its operation. Define $\alpha,\beta\in\operatorname{Homeo}^{+}({\mathbb{R}})$ by \[ \alpha(t)=t+\frac12,\qquad \beta(t)= \begin{cases} t,&t\leq0,\\ \dfrac{t}{1-t},&0\leq t\leq\frac12,\\ 3-\dfrac1t,&\frac12\leq t\leq1,\\ t+1,&t\geq1. \end{cases} \] The formulas for $\beta$ agree at the shared endpoints $0,\frac12,1$. Let \[ G=\langle\alpha,\beta\rangle\leq\operatorname{Homeo}^{+}({\mathbb{R}}). \] Regard $G$ as a discrete group. It is amenable if there exists a finitely additive function $\mu:\mathcal P(G)\to[0,1]$ such that $\mu(G)=1$ and $\mu(gA)=\mu(A)$ for every $g\in G$ and every subset $A\subseteq G$, where $gA=\{ga:a\in A\}$.

Precise yes-or-no question. Is $G$ nonamenable; that is, does no such $\mu$ exist?

For the accompanying comparison, Thompson's group $F$ denotes the group of increasing piecewise-linear homeomorphisms of $[0,1]$ with finitely many breakpoints, all dyadic rational, and slopes in $\{2^k:k\in{\mathbb{Z}}\}$. A dyadic rational is a number $m/2^n$, where $m\in{\mathbb{Z}}$ and $n$ is a nonnegative integer.

Unresolved qualifications of the source question. The record describes $G$ as a subgroup of Monod's group, but supplies no definition of that ambient group. The explicit generators nevertheless determine $G$ in $\operatorname{Homeo}^{+}({\mathbb{R}})$. The original question also asks whether this is the simplest nonamenable example going slightly beyond $F$, and why the coefficient $1/2$ is important. No criterion of simplicity, precise relationship to $F$, or comparison obtained by changing $1/2$ is specified. Thus these accompanying requests remain qualitative questions requiring clarification; they do not assert a specified embedding, extension, or other relationship between $F$ and $G$. The number $1/2$ occurs both as the translation amount in $\alpha$ and as a breakpoint in $\beta$; the record does not specify a parameterized replacement of these occurrences.

Short English statement

Determine whether the explicitly generated group $G=\langle\alpha,\beta\rangle$, considered as a discrete group, is nonamenable. The source also asks why $1/2$ matters and whether this is the simplest example slightly beyond Thompson's group $F$, but supplies no precise comparison or simplicity criterion.

Sources

• Local UnsolvedMath registry, permanent ID $30003851$, source problem code catalogue entry, title: "A Nonamenable Extension Beyond Thompson's Group." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2018; no individual proposer is identified.
• Supplied source citation: Cohomological and Metric Properties of Groups of Homeomorphisms of R (2019), Oberwolfach Reports, pp. 1579--1633, DOI 10.4171/owr/2018/26. https://doi.org/10.4171/owr/2018/26. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: "Oberwolfach Report 26/2018." https://ems.press/content/serial-article-files/46748?nt=1.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and describes the questions about the simplest nonamenable example beyond $F$ and the importance of $1/2$ as unresolved. This is supplied dated curation, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement carries the registry label corrected_verified, with a claim of manual semantic, typographic, and LaTeX review. This records the supplied review claim; it does not establish mathematical correctness or independent source verification for this draft.