Boundaries of Groups and Kleinian Groups — Problem 46

Definitions and mathematical statement

In the CAT(0) setting suggested by the supplied summary, let $\partial_\infty X$ be the visual boundary, whose points are asymptotic classes of geodesic rays. Each $g\in G=\operatorname{Isom}(X)$ induces a boundary map by \[ g\cdot[r]=[g\circ r]. \] The source asks whether this boundary action is "nice" with respect to "the metrics in the previous remark".

That preceding remark and its metrics are absent from the record. No boundary metric or class of metrics, basepoint or parameter dependence, or precise regularity property is given. For example, continuity, Lipschitz control, quasisymmetry, and uniform bounds over $g\in G$ are distinct possible requirements; none is selected by the word "nice". The ambient hypotheses on $X$ are likewise not reproduced. Thus the induced action can be described, but its requested metric property cannot be stated as a definite quantified conjecture from this source fragment.

Short English statement

Determine the regularity of the isometry-group action on a CAT(0) boundary for an unspecified earlier family of boundary metrics. Both the metrics and the meaning of "nice" are missing.

Sources

• Local UnsolvedMath registry, permanent ID 6200046, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 46"; proposed by Bruce Kleiner, 2005. Registry status open, classification OPEN-TRIAGE.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 46, PDF page 14, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. The referenced previous remark is not included, and no PDF was accessed here.
• The supplied summary calls the question open and emphasizes that CAT(0) boundary metric choices are not canonical. It supplies no metric definition or precise later reference.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the source is publicly author-hosted.