Boundaries of Groups and Kleinian Groups — Problem 80

Definitions and mathematical statement

Let $\mathbb H_{\mathbb C}^2$ be complex hyperbolic two-space, whose holomorphic isometry group is $\operatorname{PU}(2,1)$ and whose visual boundary is topologically $S^3$. For a discrete subgroup $G\le\operatorname{PU}(2,1)$, its limit set $\Lambda(G)\subseteq\partial_\infty\mathbb H_{\mathbb C}^2$ is the set of boundary accumulation points of an orbit $Gx$. Use the usual convex-cocompact condition: the quotient by $G$ of the closed geodesic convex hull of $\Lambda(G)$ in $\mathbb H_{\mathbb C}^2$ is compact.

Let $\mathcal C$ be the standard Sierpiński carpet, obtained from a closed square by repeatedly deleting the open middle square of the subdivision of each remaining square into nine equal squares. Does there exist a convex-cocompact discrete subgroup $G\le\operatorname{PU}(2,1)$ such that \[ \Lambda(G)\cong\mathcal C? \] The equivalence requested is a homeomorphism of topological spaces. No torsion-free condition, prescribed embedding of the carpet in $S^3$, or metric equivalence is added.

The supplied summary notes carpet limit sets in real-hyperbolic Kleinian settings and says no complex-hyperbolic construction was verified in its review. The real-hyperbolic examples are not asserted to yield the requested subgroup. The registry's open label and the 2005 problem are retained as dated source information, without independent current verification.

Short English statement

Can a convex-cocompact discrete group of complex hyperbolic two-space have a limit set homeomorphic to the Sierpiński carpet? The supplied record labels this open and gives no verified construction.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6200080$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Misha Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 80, PDF page 20: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29; current status marked as needing review.
• The summary mentions real-hyperbolic carpet examples without separate original references and supplies no complex-hyperbolic realization or nonexistence theorem.