Boundaries of Groups and Kleinian Groups — Problem 23

Definitions and mathematical statement

A $(\lambda,\varepsilon)$-quasigeodesic in a metric space $X$ is a map $\gamma:I\to X$ satisfying \[ \lambda^{-1}|s-t|-\varepsilon\leq d(\gamma(s),\gamma(t)) \leq\lambda|s-t|+\varepsilon \] for $\lambda\geq1$, $\varepsilon\geq0$. A CAT(0) space is geodesic with triangles no thicker than Euclidean comparison triangles. Its visual boundary $\partial_\infty X$ consists of asymptotic geodesic rays with the visual topology; a basepoint identifies these with rays beginning there.

Topology/measure program (Danny Calegari). Define a topology on the intended set of quasigeodesics in a CAT(0) space, in the source's alternatives of proper geodesic, coarsely homogeneous, or cocompact settings, such that:

1. it is an increasing union of compact metrizable spaces;
2. it includes $\partial_\infty X$;
3. it is quasi-isometry invariant;
4. it has reasonable measure classes which are quasipreserved.

The first condition can be written $B(X)=\bigcup_{m\geq1}K_m$ with $K_m\subseteq K_{m+1}$ compact metrizable. The source does not explicitly require the final/direct-limit topology. A usual precise measure-class interpretation would ask that an induced quasi-isometry homeomorphism $F_\partial:B(X)\to B(Y)$ send a chosen measure class to the chosen class, with representatives satisfying $(F_\partial)_*\mu_X\sim\mu_Y$. This is conditional terminology: "reasonable" and "quasipreserved" are undefined.

The intended underlying set is also unresolved: finite paths, rays or lines; parameterized paths or asymptotic/bounded-distance classes; and any basepoint convention are not stated. The strength of boundary inclusion (set inclusion or topological embedding), naturality under quasi-isometries, and admissible measures must be specified. The program is not silently replaced by a topology only on Morse rays, which could omit visual directions.

As context, the source reports Bowditch--Swarup [13, 56]: a one-ended hyperbolic group has a boundary with no cut points. It also reports Swenson: if a visual-boundary point of a CAT(0) group is a cut point, an infinite torsion subgroup fixes it. These are supplied contextual claims, not extra requirements of the topology. The summary additionally mentions Croke--Kleiner and says the full program is not worked out, without precise references or a modern resolution check.

Short English statement

Build a quasi-isometry-invariant quasigeodesic space for CAT(0) geometry, exhausted by compact metrizable pieces, containing the visual boundary, and carrying natural quasipreserved measure classes. The path equivalence, topology, inclusion, and measure requirements are not defined fully. Contextual cut-point theorems are preserved separately.

Sources

• Local UnsolvedMath registry, permanent ID 6200023, source code catalogue entry, "Boundaries of Groups and Kleinian Groups --- Problem 23"; proposer: Danny Calegari, proposed year 2005. Status open, classification OPEN-TRIAGE, updated 2026-08-06.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 23, PDF page 8, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Reported PDF-text-normalized extraction and access: 2026-07-29. Current status NEEDS_REVIEW; the source labels it a problem. Rights: public author-hosted PDF; statement-reuse terms NEEDS_REVIEW.
• Contextual source attributions: Brian Bowditch and Gadde Swarup, source labels [13, 56] without extracted bibliography; Eric Swenson, cut-point/torsion-subgroup claim without a complete citation. The summary names Croke--Kleiner without identifying a result.