Boundaries of Groups and Kleinian Groups — Problem 5

Definitions and mathematical statement

In the usual finite-complex setting, let $H(K)$ denote a Charney--Davis strict hyperbolization of an admissible finite connected input complex $K$. Its negatively curved local metric gives a proper geodesic CAT$(-1)$ universal cover $\widetilde{H(K)}$. The visual boundary \[ \partial_\infty\widetilde{H(K)} \] is the space of geodesic rays modulo bounded-distance equivalence, with its visual topology. For the geometric action of $G_K=\pi_1(H(K))$, this is the corresponding hyperbolic-group boundary $\partial G_K$. This is the boundary interpretation indicated by the group-boundary source context, rather than the ordinary boundary of a manifold with boundary.

Benakli's question asks what can be said about the boundaries produced by this construction. A possible precise range object is \[ \{[\partial_\infty\widetilde{H(K)}]:K\text{ in the intended admissible class}\}, \] where brackets may denote homeomorphism type if topological classification is intended. The record does not specify that input class, the role of dimension or links, or whether the requested information concerns topology, local structure, metric geometry, or a different equivalence relation. It therefore states a boundary-description programme, not a unique conjectured classification theorem.

The supplied research summary reports the study ongoing and no complete description located. Its references to negatively curved and locally CAT$(0)$ complexes are not used to conflate the local metric on the hyperbolized quotient with the global CAT$(-1)$ metric on its universal cover. No unsupported boundary classification or example is supplied.

Short English statement

Describe the group/visual boundaries arising from Charney--Davis strict hyperbolization. The admissible inputs and the desired topological or metric classification are not specified by the extracted question.

Sources

• Local UnsolvedMath registry, permanent ID $6200005$, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups --- Problem 5." Proposer Nadia Benakli, 2005.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 5, recorded PDF page 3; https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Recorded access 2026-07-29; pdf-text-normalized extraction.
• The statement cites Charney and Davis as reference [18], but the record does not supply that bibliography entry. The OPEN-TRIAGE research summary adds no precise classification reference.