Boundaries of Groups and Kleinian Groups — Problem 24

Definitions and mathematical statement

The supplied record is largely definitional and contains no explicit final question. It begins with the assertion that every CAT(0) group has no infinite torsion subgroup. If this is intended as the problem's conjectural target, its quantified form is \[ \forall G\text{ CAT(0)}\ \forall H\le G:\quad \left(\forall h\in H\ \exists m\ge1:\ h^m=e\right)\Longrightarrow |H|<\infty. \] Here a CAT(0) group conventionally acts properly and cocompactly by isometries on an appropriate proper CAT(0) space, and a torsion subgroup means all its elements have finite order. The record does not distinguish a proposed assertion from background fact, so this claim is not certified as a theorem here.

Retaining the supplied definitions, a Z-structure is a pair $(\widetilde X,Z)$ with the following properties. The space $\widetilde X$ is a compact Euclidean retract, described in the source as a compact space embedded as a retract in some $\mathbb R^n$. The set $Z$ is a closed Z-set, homotopically negligible in the sense that for every open $U\subseteq\widetilde X$, \[ U\setminus Z\hookrightarrow U \] is a homotopy equivalence. The complement $X=\widetilde X\setminus Z$ admits a covering-space action of $G$ with compact quotient $X/G$. Finally the translates of each compact $K\subset X$ form a null family: for every $\varepsilon>0$ only finitely many translates $gK$ have diameter greater than $\varepsilon$, in a compatible metric on $\widetilde X$. A space $Z$ occurring in such a pair is a Z-structure boundary of $G$.

The displayed inclusion corrects the extraction's malformed "$U\ \mathbb Z$ in $U$" to the complement inclusion indicated by homotopical negligibility. The Euclidean-retract wording does not explicitly distinguish a retract of the entire ambient Euclidean space from a neighborhood retract, and should be checked in the original.

The source then calls $Z$ an EZ-boundary, denoted $\partial_{\mathrm{EZ}}G$, when the action on $X$ "extends" to a topological action on $Z$. The usual meaningful extension condition is an action by homeomorphisms on the whole compactification $\widetilde X$, agreeing with the given action on $X$ and preserving $Z$. Merely giving a separate action on the disjoint subset $Z$ would not express continuity across the boundary. This missing compatibility must be confirmed.

The registry is labeled open, but its research summary classifies the material as background/literature survey and speaks broadly of classifying Z-structures without stating a particular classification question. No new classification target is invented. It attributes related theory to Farrell--Lafont and Bestvina--Mess without precise citations. The intended question and the status of the opening torsion assertion remain unresolved from this excerpt.

Short English statement

The record gives definitions of Z-structures and EZ-boundaries and an opening assertion about torsion subgroups of CAT(0) groups, but no explicit question. The boundary-action compatibility and one extracted inclusion need clarification before a specific research statement can be identified.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6200024$, source code catalogue entry; complete supplied statement, background, and research summary, updated 2026-08-06. Attributed to Eric Swenson, 2005.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 24, PDF page 8: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29; current status marked as needing review.
• T. Farrell and J. Lafont are credited in the statement for EZ-boundaries; the summary also names Farrell--Lafont and Bestvina--Mess without full publication details.