Finiteness Properties of Hyperbolic Tessellation Groups

Definitions and mathematical statement

Let $\mathbb H^2$ denote the hyperbolic plane, and let $\Delta$ be a tessellation of $\mathbb H^2$. The supplied record imposes no further conditions on the tessellation. In the usual countable-set convention, the finitary symmetric group is \[ S_\infty=\{\sigma\in\operatorname{Sym}(\Omega):\{x\in\Omega:\sigma(x)\ne x\}\text{ is finite}\}, \] where $\Omega$ is a countably infinite set. The record does not specify the underlying set or its relation to $\Delta$.

Thompson's group $V$ is the group of homeomorphisms of $\{0,1\}^{{\mathbb{N}}}$ given by finite binary prefix replacements: partition the space into finitely many cylinder sets $u_i\{0,1\}^{{\mathbb{N}}}$, choose another such partition $v_i\{0,1\}^{{\mathbb{N}}}$, and map $u_iw$ to $v_iw$.

Write $G=\phi(\Delta)$. The construction $\phi(\Delta)$ is not defined in the supplied record. Its quotient notation presupposes a specified normal subgroup of $G$ identified with $S_\infty$. Subject to that missing construction and identification, the hypothesis is \[ G/S_\infty\cong V. \] Question. For every tessellation covered by the intended construction and satisfying this hypothesis, do $G$ and $G/S_\infty$ have the same finiteness properties?

The expression "same finiteness properties" is not specified in the record. For example, a group $H$ is of type $F_n$, for an integer $n\geq0$, if it admits a classifying CW complex $K(H,1)$ with finite $n$-skeleton. If the intended properties are precisely these geometric finiteness properties, the question becomes whether \[ \forall n\in{\mathbb{Z}}_{\geq0},\qquad G\text{ is of type }F_n \ \Longleftrightarrow\ G/S_\infty\text{ is of type }F_n. \] This is a conditional interpretation, not a resolution of the source's terminology. Homological finiteness properties, such as type $FP_n$ with specified coefficients, may also be intended. The missing definition of $\phi(\Delta)$, the admissible tessellations, the normal subgroup identification, and the intended family of finiteness properties require clarification before a fully determined statement is available.

Short English statement

The supplied question asks whether a group associated with a hyperbolic tessellation has the same finiteness properties as its quotient by finitary permutations, assuming that quotient is Thompson's group $V$. The construction of the group and the intended meaning of finiteness properties remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003834$, source problem code catalogue entry, titled "Finiteness Properties of Hyperbolic Tessellation Groups," in the collection "Oberwolfach Reports Open Problems." 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. The citation's 2019 label and the 2018 year in the DOI are retained as supplied.
• Supplied literature source labeled "Oberwolfach Report 2019": https://ems.press/content/serial-article-files/46748?nt=1.
• Supplied literature source attributed to Farley--Hughes: https://doi.org/10.4171/JCA/91. No article title or pinpoint supporting this question is supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the question open, describes equality of finiteness properties as open in general, and reports known special Euclidean and Houghton-type cases. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.