Functional Characterization of Thompson Group Amenability

Definitions and mathematical statement

Let $F$ denote Thompson's group: the group, under composition, of increasing piecewise linear homeomorphisms of $[0,1]$ with finitely many breakpoints, all dyadic rational, and slopes in $\{2^k:k\in{\mathbb{Z}}\}$. Here a dyadic rational is a number $m/2^r$ with $m\in{\mathbb{Z}}$ and $r\in{\mathbb{Z}}_{\geq 0}$. Say that $F$ is amenable if there exists a finitely additive probability measure $\mu:\mathcal P(F)\to[0,1]$ such that $\mu(gE)=\mu(E)$ for every $g\in F$ and every $E\subseteq F$.

Let $\mathcal T$ be the set of finite formal terms generated by the constant symbol $1$ and the binary operation symbol $\star$: thus $1\in\mathcal T$, and $A\star B\in\mathcal T$ whenever $A,B\in\mathcal T$. For this draft, interpret being related by the displayed identity as the least congruence $\sim$ on $\mathcal T$ containing \[ A\star(B\star C)\sim (A\star B)\star(A\star C) \qquad(A,B,C\in\mathcal T). \] A congruence here is an equivalence relation preserved by $\star$ in both arguments. This specifies closure under replacements inside larger terms; the supplied record does not explicitly state that closure convention.

Write ${\mathbb{N}}$ for the natural-number domain in the supplied record, whose starting index is unspecified. For a strictly increasing nonidentity function $f:{\mathbb{N}}\to{\mathbb{N}}$, interpret its critical point as \[ \operatorname{crit}(f)=\min\{n\in{\mathbb{N}}:f(n)\ne n\}. \] This is the conventional first-moved-point interpretation; the supplied record does not define $\operatorname{crit}$, so this interpretation requires confirmation against the report.

With these qualifications, the proposed equivalence is \[ F\text{ is amenable} \quad\Longleftrightarrow\quad \text{there exists a family }(f_T)_{T\in\mathcal T}\text{ satisfying the conditions below}. \] For every $T\in\mathcal T$, the function $f_T:{\mathbb{N}}\to{\mathbb{N}}$ is strictly increasing and is not the identity. The remaining conditions are \begin{align*} S\sim T&\ \Longrightarrow\ f_S=f_T &&(S,T\in\mathcal T),\\ \operatorname{crit}(f_{A\star B})&=f_A\left(\operatorname{crit}(f_B)\right) &&(A,B\in\mathcal T),\\ \{\operatorname{crit}(f_T):T\in\mathcal T\}&={\mathbb{N}}. \end{align*} The last condition means that every natural number occurs as the critical point of at least one member of the family. The equivalence is posed as a question, not asserted as a theorem.

Short English statement

Does amenability of Thompson's group $F$ hold exactly when there is a family of strictly increasing nonidentity maps of ${\mathbb{N}}$, indexed by variable-free terms, respecting the displayed distributive identity and critical-point recursion, with every natural number occurring as a critical point? The meanings of critical point, identity-generated equivalence, and the starting index of ${\mathbb{N}}$ require source confirmation.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30001683$, title "Functional Characterization of Thompson Group Amenability," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2011; no proposer is named.
• Supplied report citation: Set Theory (2011), Oberwolfach Reports, pp. 85--140, DOI 10.4171/owr/2011/02. https://doi.org/10.4171/owr/2011/02. This is also the supplied literature source labelled "Oberwolfach problem statement."
• Supplied literature source labelled "current amenability survey": https://arxiv.org/abs/2305.07113. No author or title is supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the proposed equivalence as conjectural alongside the unresolved amenability problem for $F$. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports that the official report was checked and the properties defining the function system were restored. These are supplied provenance claims; this draft does not independently verify them. The shorter original statement mentions only the equivalence and the term-indexed system; the detailed conditions above come from the supplied expanded statement.