Virtual Fibrations with Precisely Controlled Finiteness Properties

Definitions and mathematical statement

For a group $G$ and an integer $m\geq 0$, say that $G$ is of type $F_m$ if it admits a classifying CW complex $K(G,1)$ with finite $m$-skeleton. Here a $K(G,1)$ is a connected CW complex with fundamental group isomorphic to $G$ and contractible universal cover. Thus the requested distinction concerns homotopical finiteness properties.

The supplied background places the question in the setting of lattices $\Lambda$ in $\operatorname{SO}(2n,1)$. A lattice in a Lie group is a discrete subgroup whose quotient has finite invariant volume. However, the supplied statement does not specify $\Lambda$, the range of $n$, any further lattice hypotheses, or the quantification over these parameters. With the standard definition above, the expression $F_{n-1}$ requires an integer $n\geq 1$; this is a requirement for interpreting the notation, not a recovered source hypothesis.

For a specified pair $(n,\Lambda)$ within the intended but incompletely supplied scope, the extracted question asks whether \[ \exists\,\Lambda_0<\Lambda\quad \exists\,\phi:\Lambda_0\longrightarrow{\mathbb{Z}} \quad\text{such that}\quad [\Lambda:\Lambda_0]<\infty,\qquad \phi\text{ is a group homomorphism},\qquad \ker\phi\text{ is of type }F_{n-1}\text{ but not of type }F_n. \] Here $[\Lambda:\Lambda_0]$ is the number of cosets of $\Lambda_0$ in $\Lambda$, and $\ker\phi=\{g\in\Lambda_0:\phi(g)=0\}$. The subgroup symbol $<$ is retained from the source; the record does not explain whether properness is intended. The displayed target imposes no explicit nontriviality or surjectivity condition on $\phi$. Such a condition must not be inferred solely from the terminology of algebraic fibrations. The missing parameter scope prevents this display from being a fully specified global conjecture.

Short English statement

For the intended lattice $\Lambda$, the question asks for a finite-index subgroup admitting a homomorphism to ${\mathbb{Z}}$ whose kernel has a classifying space with finite $(n-1)$-skeleton but none with finite $n$-skeleton. The supplied record leaves the dimension range and precise lattice hypotheses and quantifiers unresolved.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30005033$, source problem code catalogue entry, title Virtual Fibrations with Precisely Controlled Finiteness Properties, in the Oberwolfach Reports Open Problems collection. The registry records proposed year 2022 and status open; no individual proposer is supplied.
• Supplied report citation: Geometric Structures in Group Theory (2023), Oberwolfach Reports, pp. 517--576, DOI 10.4171/owr/2022/11. https://doi.org/10.4171/owr/2022/11. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Improved algebraic fibrings. https://arxiv.org/abs/2112.00397. The supplied excerpt attributes supporting evidence to Fisher's recent proof, cited there as [2], of a conjecture of Kielak, cited there as [7, 8], about finiteness properties of kernels of virtual algebraic fibrations of RFRS groups. The record does not supply the bibliography identifying those numbered references.
• Supplied literature assessment, checked on 2026-08-22: the record labels the question open and states that RFRS theory provides homological $\mathrm{FP}_{n-1}$ but not $\mathrm{FP}_n$ kernels in all dimensions, while the requested homotopical $F_{n-1}$ but not $F_n$ result for lattices in $\operatorname{SO}(2n,1)$ remains a separate unresolved question. This is supplied dated curation derived from the OWR material, not an independently verified assessment. The supplied checked-source links are https://doi.org/10.4171/owr/2022/11 and https://arxiv.org/abs/2112.00397.
• The registry assigns statement status corrected_verified and reports that the official report was checked directly and that the cleaned statement preserves the complete target while restoring definitions and removing extraction artifacts or source-dependent wording. These are supplied verification claims; no independent source visit or mathematical verification is asserted here.