Finitely Presented Piecewise-Projective Groups

Definitions and mathematical statement

The supplied registry record combines two independent questions and marks its statement as unrecoverable; it does not promote a verified clean statement. The formulations below preserve those two questions provisionally, with the unresolved terminology stated explicitly.

A group is finitely presented if it admits a presentation with finitely many generators and finitely many defining relations. A group $G$ has type $F_\infty$, written $F^{\infty}$ in the source, if there exists an Eilenberg--Mac Lane CW complex $K(G,1)$ whose $n$-skeleton is finite for every integer $n\geq 0$. An embedding of groups means an injective group homomorphism.

For a homeomorphism $g$ of an interval $I$, define its moved set by \[ M(g)=\{x\in I:g(x)\neq x\}. \] The usual one-bump condition is that $M(g)$ is a single nonempty interval; agreement with the source's precise convention requires review. The source writes $\mathrm{PLo}(I)$, customarily suggesting orientation-preserving piecewise-linear homeomorphisms of an interval, but does not specify $I$, its boundary conditions, or the precise piecewise-linearity convention. Its title refers instead to piecewise-projective groups. These choices cannot be resolved from the supplied evidence. Write $\operatorname{Fast}(S)$ solely as a placeholder for the source's undefined condition that $S$ is a fast set; no mathematical definition of this predicate is supplied here.

Question 1, conditional on recovering these conventions. Is it true that every finitely presented subgroup $G\leq\mathrm{PLo}(I)$ which admits a generating set $S\subseteq G$ satisfying $\operatorname{Fast}(S)$, with every member of $S$ a one-bump element, has type $F_\infty$? Equivalently, the intended universal assertion appears to be \[ \forall G\leq\mathrm{PLo}(I),\quad \left[ \begin{gathered} G\text{ is finitely presented},\\ \exists S\subseteq G:\ G=\langle S\rangle,\quad \operatorname{Fast}(S),\\ \forall s\in S,\ M(s)\text{ is a single nonempty interval} \end{gathered} \right] \Longrightarrow G\text{ has type }F_\infty. \] This is an interpretation of the supplied wording, not a verified recovery of the numbered source question. The record does not explicitly require the fast generating set itself to be finite, so that restriction has not been inserted.

Question 2. Let \[ \Gamma=\left\langle a,b,c,d\ \middle|\ b^a=b^2,\ c^b=c^2,\ d^c=d^2,\ a^d=a^2\right\rangle \] be the Higman group appearing in the record, using the conjugation convention $x^y=y^{-1}xy$. Does there exist a Lodha--Moore group $L$ and an injective homomorphism $\Gamma\hookrightarrow L$? The source also asks the weaker target question: does an injective homomorphism \[ \Gamma\hookrightarrow H({\mathbb{R}}) \] exist, where $H({\mathbb{R}})$ denotes Monod's group named in the source? The record supplies neither a definition of the allowed Lodha--Moore groups nor a definition of $H({\mathbb{R}})$; their exact meanings and the claimed weaker-target relationship require source verification.

Short English statement

The record merges two questions: whether every finitely presented subgroup of the stated interval homeomorphism group that has fast one-bump generators is of type $F_\infty$, and whether the displayed Higman group embeds into a Lodha--Moore group, or at least into Monod's $H({\mathbb{R}})$. Essential conventions and target-group definitions are missing. These are provisional statement-only formulations requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent numeric ID $30003852$, exact source problem code catalogue entry, and title Finitely Presented Piecewise-Projective Groups. The record lists proposed year 2018 and supplies no named proposer. Its statement-verification field reports that the extraction merges two independent numbered questions, leaves the original record visible for provenance, and promotes no separate clean statement; the statement status is unrecoverable.
• 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 Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No exact page or separate question numbers for these two questions are supplied.
• Supplied literature source: Oberwolfach Report 26/2018. https://ems.press/content/serial-article-files/46748?nt=1.
• Supplied literature source: Lodha, A nonamenable type $F_\infty$ group. https://ems.press/journals/ggd/articles/13662.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the questions open and states that known type $F_\infty$ results for particular Lodha--Moore groups settle neither question. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims. No source was visited for this draft.