Borel–Serre-Type Compactifications for General Lattices

Definitions and mathematical statement

Let $G$ be a semisimple Lie group and let $\Gamma\leq G$ be a lattice, meaning a discrete subgroup for which $G/\Gamma$ has finite $G$-invariant volume. Regard $\Gamma$ as a discrete group. A model for the classifying space for proper $\Gamma$-actions, denoted $\underline{E}\Gamma$, is a $\Gamma$-CW complex $Y$ such that, for every subgroup $H\leq\Gamma$, \[ Y^H=\{y\in Y:hy=y\text{ for every }h\in H\} \quad\text{satisfies}\quad \begin{cases} Y^H\text{ is contractible},&H\text{ is finite},\\ Y^H=\varnothing,&H\text{ is infinite}. \end{cases} \] Use cofinite here to mean that $Y$ has finitely many $\Gamma$-orbits of cells. This convention requires confirmation against the source's intended use of the term.

Question. For every lattice $\Gamma$ in a semisimple Lie group $G$, does there exist an analogue of the Borel--Serre compactification that provides a cofinite model $Y$ for $\underline{E}\Gamma$? In particular, the requested construction must satisfy the fixed-point conditions above and have finitely many $\Gamma$-orbits of cells.

Unresolved qualifications. The supplied record does not specify what additional geometric, boundary, functoriality, or compactification properties constitute an "analogue of the Borel--Serre compactification." Consequently, the existence of a cofinite model alone does not fully specify the requested target. Nor does the record state conventions concerning connectedness, the centre, or compact factors of $G$, or impose irreducibility, arithmeticity, torsion-freeness, or nonuniformity on $\Gamma$. No such restrictions are added here. The intended scope of the universal quantifier over semisimple Lie groups and the intended meaning of the compactification analogue require mathematical review.

Short English statement

The question asks whether every lattice in a semisimple Lie group admits a Borel--Serre-type construction giving a classifying space for proper actions with finitely many cell orbits. Such a space must have contractible fixed-point sets for finite subgroups and no fixed points for infinite subgroups. The required extra properties of the compactification analogue and the precise scope of the Lie groups are unspecified. The supplied partially_solved label is a dated curation assessment, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: permanent ID $30001807$, title Borel--Serre-Type Compactifications for General Lattices, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns the year 2011 and supplies no individual proposer.
• Supplied authoritative report reference: Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences (2011), Oberwolfach Reports, pp. 1637--1708, DOI 10.4171/owr/2011/30. https://doi.org/10.4171/owr/2011/30. This is also the supplied literature-source link labelled "original Oberwolfach report." No problem-specific page or named contributor is supplied.
• Supplied literature source labelled "Kang--cofinite classifying spaces": https://doi.org/10.4134/CKMS.c160222. The record supplies no full bibliographic citation or theorem number.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem partially solved and attributes cofinite models for arithmetic and higher-rank irreducible lattices to Borel--Serre, and rank-one analogues for general lattices to Kang. It further states that a single canonical Borel--Serre-type construction in the full generality of the question is not established. These are supplied dated curation claims, not independently verified here; their hypotheses and time-sensitive status require independent checking.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not a source consultation performed for this draft.