Boundaries of Groups and Kleinian Groups — Problem 82

Definitions and mathematical statement

Let $\mathbb H^4$ be real hyperbolic four-space, with boundary $S^3$ carrying the standard round metric. For a discrete group $G\leq\operatorname{Isom}(\mathbb H^4)$, its limit set $\Lambda(G)\subset S^3$ is the accumulation set of an orbit $Gx$ in the boundary. The group is convex cocompact if the quotient of the convex hull of $\Lambda(G)$ by $G$ is compact. Consider non-elementary groups that are free of finite rank at least two.

The question asks whether \[ \forall\varepsilon>0\ \exists G\text{ free and convex cocompact in }\operatorname{Isom}(\mathbb H^4), \qquad \dim_H\Lambda(G)>3-\varepsilon. \] Hausdorff dimension here is for the round boundary metric; it is at most three. The source also asks the more restrictive Schottky version. If Schottky means a classical round-ball pairing construction, it is a specific subclass and should not be silently identified with all free convex-cocompact groups. The record does not give its Schottky convention.

No fixed rank is specified; in the displayed existence formulation the rank may depend on $\varepsilon$. No uniform geometric constants or single group attaining dimension three are requested. A fixed-rank version would be a separate strengthening requiring source clarification.

Short English statement

Can free convex-cocompact groups in real hyperbolic four-space have limit sets with Hausdorff dimension arbitrarily close to the ambient boundary dimension three? Ask also for the source's Schottky subclass.

Sources

• Local UnsolvedMath registry, permanent ID 6200082, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 82"; proposed by Lewis Bowen, 2005. Registry status open, classification OPEN-TRIAGE.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 82, PDF page 21, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. No PDF was accessed here.
• The supplied summary reports analogous free convex-cocompact limit sets approaching dimension two in $\mathbb H^3$, without a precise construction or citation, and says it verified no analogous dimension-three construction in $\mathbb H^4$. This is a supplied open/triage assessment, not an independent status check.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the document is publicly author-hosted.