Subgroups of Piecewise-Projective Homeomorphism Groups

Definitions and mathematical statement

The supplied record does not determine a precise mathematical proposition. Its cleaned formulation asks whether Thompson's group $F$ is the only "structurally significant" subgroup of $\operatorname{PL}_o(I)$, described there as the group of orientation-preserving piecewise-linear homeomorphisms of an interval. Its original formulation instead asks whether $F$ is the only "interesting" subgroup of a group described as consisting of piecewise-projective homeomorphisms, using the same notation. The supplied title also refers to piecewise-projective homeomorphisms. These descriptions do not specify the same ambient group.

For reference, a standard concrete realization of Thompson's group is \[ F=\left\{f\colon[0,1]\longrightarrow[0,1]\;\middle|\; \begin{array}{l} f\text{ is an increasing piecewise-linear homeomorphism},\\ f\text{ has finitely many breakpoints, all in }\mathbb Z[1/2],\\ \text{every linear piece has slope in }\{2^k:k\in\mathbb Z\} \end{array}\right\}, \] with group operation composition, where \[ \mathbb Z[1/2]=\{m/2^n:m\in\mathbb Z,\ n\in\mathbb Z_{\geq 0}\}. \] In this realization, every element fixes $0$ and $1$. Under the convention of finitely many linear pieces, the group of all increasing piecewise-linear homeomorphisms of $[0,1]$ contains this realization of $F$. This supplies a standard reference model; the record does not explicitly specify that $I=[0,1]$, its endpoint conditions, or its finiteness convention. Piecewise-projective maps instead have pieces given by fractional linear transformations, so the original wording cannot silently be replaced by the piecewise-linear interpretation.

Unresolved statement. Determine the intended ambient group $G$, the meaning of "interesting" or "structurally significant" for a subgroup $H\leq G$, and the intended meaning of uniqueness. In particular, the record does not specify whether uniqueness means equality with a distinguished embedded copy of $F$, conjugacy to such a copy, or isomorphism to $F$. Without these specifications, no definite quantified uniqueness assertion can be extracted. The supplied label open describes the record's curation status; it does not establish a well-defined open conjecture.

Short English statement

The record asks whether Thompson's group $F$ is the only subgroup of a certain homeomorphism group deserving an unspecified structural distinction. A precise conjecture requires clarification of that distinction, the ambient piecewise-linear versus piecewise-projective group, the interval conventions, and the meaning of uniqueness.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003842$, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is $2018$.
• Supplied report 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 documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Brin": https://arxiv.org/abs/math/0501085. No further bibliographic details or supporting passage are supplied.
• Supplied literature source labeled "Oberwolfach Report 2019": https://ems.press/content/serial-article-files/46748.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, observes that "interesting subgroup" is not a precise theorem, and describes the subgroup structure of $\operatorname{PL}_o(I)$ as poorly understood in the primary literature. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The record labels its cleaned statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting preservation of the mathematical target. This is supplied review metadata; it does not resolve the formulation conflicts identified here. No authoritative source passage resolving those conflicts is supplied.