Quasi-Retracts of Thompson's Group $F$

Definitions and mathematical statement

Let $D={\mathbb{Z}}[1/2]$. Thompson's group $F$ is the group of increasing piecewise linear homeomorphisms of $[0,1]$ with finitely many breakpoints, all in $D$, and slopes in $\{2^k:k\in{\mathbb{Z}}\}$. Thompson's group $T$ is the group of orientation-preserving piecewise linear homeomorphisms of ${\mathbb{R}}/{\mathbb{Z}}$ with finitely many breakpoints in $D/{\mathbb{Z}}$, slopes in $\{2^k:k\in{\mathbb{Z}}\}$, and preserving $D/{\mathbb{Z}}$. The notation $F\times F$ denotes the direct product.

The supplied source asks whether $F$ has any "interesting proper nonabelian quasi-retracts," and specifically asks:

1. Is $F$ a quasi-retract of $T$?
2. Is $F\times F$ a quasi-retract of $F$?

The supplied record does not define "quasi-retract," specify the relevant embeddings, or give a mathematical criterion for "interesting." Consequently, the following metric formulation is conditional on a convention that requires confirmation against the report.

For a finitely generated group $G$ with word metric $d_G$ and a finitely generated subgroup $H\leq G$ with its own word metric $d_H$, a coarse Lipschitz retraction means a map $r:G\to H$ and constants $L\geq1$, $A,B\geq0$ such that \[ d_H(r(x),r(y))\leq Ld_G(x,y)+A \quad\text{for all }x,y\in G, \qquad d_H(r(h),h)\leq B \quad\text{for all }h\in H. \] Under the convention that existence of such a map makes $H$ a quasi-retract of $G$, the two questions ask for such retractions for the respective pairs. Natural candidate embeddings are $F\hookrightarrow T$, obtained by identifying the endpoints of $[0,1]$, and $F\times F\hookrightarrow F$, obtained by letting the two factors act on $[0,1/2]$ and $[1/2,1]$ after affine rescaling. Neither this convention nor these particular embeddings are specified in the supplied record; they are proposed interpretations, not confirmed source hypotheses. The broader question about "interesting" proper nonabelian quasi-retracts remains an informal request without a specified class of subgroups or a criterion for interest.

Short English statement

The source asks whether Thompson's group $F$ has interesting proper nonabelian quasi-retracts, and separately whether $F$ is a quasi-retract of $T$ and whether $F\times F$ is a quasi-retract of $F$. A precise metric interpretation is supplied conditionally because the record does not define quasi-retract or specify embeddings. The supplied open status requires independent verification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003848$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• 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.
• Supplied literature source: Oberwolfach Report 26/2018. https://ems.press/content/serial-article-files/46748?nt=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that the report poses both quasi-retract questions as open, and reports that no later primary resolution of either question was found. This is supplied dated curation, not an independently verified literature assessment; neither linked source was visited for this draft.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial assertions, not independent mathematical verification of this draft.