Finiteness and Rigidity of Fast Bump Groups

Definitions and mathematical statement

A group $G$ is of type $F_\infty$ if there exists a connected CW complex $X$ with $\pi_1(X)\cong G$, contractible universal cover, and finite $k$-skeleton for every integer $k\geq 0$.

For each integer $n>1$, let $\mathcal C_n$ provisionally denote the class intended by the source: groups generated by $n$ fast one-bump functions, excluding nontrivial direct sums and wreath products. This notation records an unresolved source qualification: the supplied record does not define the ambient homeomorphism group, "one-bump," or "fast," and does not specify precisely which direct-sum and wreath-product decompositions are excluded. Consequently, $\mathcal C_n$ cannot be defined unambiguously from the supplied record alone. Generation by $n$ functions is not stated to mean that $n$ is the minimum number of generators.

The finiteness question is, conditional on specifying this class, \[ \forall n\in{\mathbb{Z}},\quad n>1\ \Longrightarrow\ \forall G\in\mathcal C_n,\quad G\text{ is of type }F_\infty. \] The source also asks whether all such groups are isomorphic. Under the reading that comparisons are made for each fixed $n$, this is \[ \forall n\in{\mathbb{Z}},\quad n>1\ \Longrightarrow\ \forall G,H\in\mathcal C_n,\quad G\cong H. \] The supplied wording does not explicitly settle whether the isomorphism question instead compares groups obtained for different values of $n$. These are questions, not asserted conclusions, and the displayed formulations remain conditional on resolving the missing definitions and exclusions.

Short English statement

For each integer $n>1$, the source asks whether groups generated by $n$ fast one-bump functions, subject to its direct-sum and wreath-product exclusions, have type $F_\infty$, and whether all such groups are isomorphic. The technical hypotheses, exclusions, and scope of the isomorphism comparison require clarification before this becomes a fully specified mathematical statement.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003846$, title "Finiteness and Rigidity of Fast Bump Groups," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2018 and names no 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.
• 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 Questions 71--72 unresolved in general, reports affirmative answers only for $n=2,3$ and conjectural behavior for $n>3$, and states that no later primary resolution was found. This is supplied dated curation, not an independently verified literature assessment. The source record itself requests independent verification of time-sensitive status and cited claims.
• The registry supplies the labels open and corrected_verified, together with a claim of manual semantic, typographic, and LaTeX review. These are supplied registry annotations; this draft does not independently verify them or the cited report.