Well-Quasi-Ordering of Thompson Subgroups

Definitions and mathematical statement

Put $I=[0,1]$. Use the standard convention that $\operatorname{PL}_o(I)$ is the group, under composition, of increasing piecewise-linear homeomorphisms $I\to I$ with finitely many linear pieces. In particular, these homeomorphisms fix both endpoints. Thompson's group $F$ is the subgroup consisting of maps whose breakpoints are dyadic rationals and whose slopes belong to $\{2^k:k\in{\mathbb{Z}}\}$.

For groups $G,H$, write $G\preccurlyeq H$ if there exists an injective group homomorphism $G\to H$. Say that $G$ and $H$ are biembeddable, and write $G\sim H$, if both $G\preccurlyeq H$ and $H\preccurlyeq G$. For an ambient group $A$, let \[ \mathcal B(A)=\{[H]_{\sim}:H\leq A\}, \qquad [G]_{\sim}\leq[H]_{\sim}\ \Longleftrightarrow\ G\preccurlyeq H. \] This defines a partial order on the biembeddability classes. A partially ordered set $(P,\leq)$ is well-quasi-ordered if every sequence $(p_n)_{n\in{\mathbb{N}}}$ in $P$ has indices $i<j$ with $p_i\leq p_j$.

Questions.

1. Is $\mathcal B(F)$ well-quasi-ordered? Equivalently, is it true that \[ \forall (H_n)_{n\in{\mathbb{N}}}\in\{H:H\leq F\}^{{\mathbb{N}}},\quad \exists i<j\quad H_i\preccurlyeq H_j? \]
2. Does $\mathcal B(\operatorname{PL}_o(I))$ fail to be well-quasi-ordered? Equivalently, does there exist a sequence $(K_n)_{n\in{\mathbb{N}}}$ of subgroups of $\operatorname{PL}_o(I)$ such that \[ \forall i<j,\quad K_i\not\preccurlyeq K_j? \]

Scope and unresolved qualifications. The supplied statement says subgroups without imposing finite generation, countability, or other subgroup restrictions; the displayed questions retain that unrestricted scope. Embeddings are interpreted here as abstract injective group homomorphisms, without an additional condition relating the actions on $I$. The source record does not explicitly define its embedding convention or $\operatorname{PL}_o(I)$; the standard conventions above require confirmation against the cited report. The second question is interpreted as asking for a family of subgroup classes witnessing failure of well-quasi-ordering. A single class alone cannot witness such failure.

Short English statement

Does every sequence of subgroups of Thompson's group $F$ contain an earlier group that embeds into a later one? Can a sequence of subgroups of $\operatorname{PL}_o([0,1])$ fail this property? The draft uses abstract group embeddings and retains the source's unrestricted subgroup quantifier. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003845$, title Well-Quasi-Ordering of Thompson Subgroups, exact source problem code catalogue entry, in Oberwolfach Reports Open Problems. The record lists proposed year 2018 and supplies no individual proposer.
• 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 background attributes the collection to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature reference: Bleak--Brin--Moore, https://doi.org/10.4171/JCA/49. The supplied assessment attributes to these authors a long well-ordered restricted family and incomparable examples; it does not specify that family's hypotheses or order type.
• Supplied literature reference labeled Oberwolfach Report 2019: https://ems.press/content/serial-article-files/46748.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. The supplied assessment labels both the full biembeddability well-quasi-ordering question and the $\operatorname{PL}_o(I)$ violation question open. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial assertions, not independent mathematical or source verification for this draft.