Cofinite Models for Universal Proper-Action Spaces

Definitions and mathematical statement

The supplied record does not specify the category of groups or recover the complete original conjecture. The following is a conditional formulation in the usual discrete-group setting; its applicability to the cited report requires review.

Let $\Gamma$ be a discrete group. A model $X=\underline{E}\Gamma$ for the universal space for proper $\Gamma$-actions is a $\Gamma$-CW complex satisfying, for every subgroup $H\leq\Gamma$, \[ X^H:=\{x\in X: hx=x\text{ for every }h\in H\} \quad\text{is}\quad \begin{cases} \text{nonempty and contractible},& H\text{ finite},\\ \varnothing,& H\text{ infinite}. \end{cases} \] The supplied record writes this space as $E\Gamma$. The condition for infinite subgroups makes explicit the proper-action requirement; the finite-subgroup condition alone does not express it.

Using the usual meaning of finite CW type, define the requested property by \[ \mathcal P(\Gamma)\;\Longleftrightarrow\; \exists\,X\text{ a model for }\underline{E}\Gamma\; \exists\,K\text{ a finite CW complex such that } \Gamma\backslash X\simeq K, \] where $\simeq$ denotes homotopy equivalence. The supplied problem asks for a characterization of the groups $\Gamma$ satisfying $\mathcal P(\Gamma)$, together with explicit models witnessing this property, especially for arithmetic groups and mapping class groups.

The record calls such a model "cofinite" by reference to finite CW type. This formulation retains that stated condition. It must be distinguished from requiring a model with finitely many $\Gamma$-orbits of cells. If that stronger requirement was intended, it gives a different formulation that the supplied evidence does not settle. The intended classes of arithmetic groups and mapping class groups are also unspecified. The supplied background describes the extraction as only the tail of an unspecified conjecture, so this restatement cannot identify the complete original conjecture.

Short English statement

Under an explicitly conditional discrete-group interpretation, characterize groups whose universal proper-action space has an orbit space homotopy equivalent to a finite CW complex, and construct explicit examples, especially for arithmetic groups and mapping class groups. The intended meaning of "cofinite," the relevant group families, and the complete original conjecture require clarification.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30001108$, source code catalogue entry, title "Cofinite Models for Universal Proper-Action Spaces," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2008.
• Supplied report citation: Mini-Workshop: Symmetric Varieties and Involutions of Algebraic Groups (2009), Oberwolfach Reports, pp. 2985--3028, DOI 10.4171/owr/2008/53. Supplied source URL and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2008/53. The supplied citation says 2009, whereas the DOI and proposed-year metadata refer to 2008; this discrepancy is unresolved.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the extraction is only the tail of an unspecified conjecture, with surrounding discussion concerning cofinite universal spaces and broad existence and classification questions. This is supplied dated curation, not an independently verified literature assessment. The linked report has not been consulted for this draft.
• The registry labels the statement corrected_verified and describes its cleanup as meaning-preserving, with no unsupported source-dependent additions. These are supplied editorial claims; they do not resolve the incomplete extraction or constitute mathematical verification here. No precise conjecture number, passage locator, or authoritative quotation establishing this formulation is supplied.