Boundaries of Groups and Kleinian Groups — Problem 8

Definitions and mathematical statement

Consider a proper geodesic polyhedral complex $X$ with a complete CAT($-1$) metric. Proper means that closed bounded sets are compact; CAT($-1$) means that triangle distances are bounded above by the corresponding distances in the hyperbolic plane of curvature $-1$.

Assume each $m$-dimensional cell is a convex hyperbolic polytope combinatorially equivalent to an $m$-cube, and faces are glued by isometries respecting their cubical face structures. These are hyperbolic cubical complexes: the cells are not ordinary Euclidean unit cubes, whose flat two-dimensional interiors would be incompatible with a strictly negative curvature upper bound.

The ideal boundary $\partial_\infty X$ is the set of geodesic rays modulo bounded-distance equivalence, with the cone topology obtained by representing rays at a common basepoint and taking uniform convergence on bounded time intervals.

Find necessary topological conditions for a compact metrizable space $Z$ to occur as \[ Z\cong\partial_\infty X \] for such an $X$, and determine how restrictive cubical cell structure is compared with general CAT($-1$) complexes. The source records obstructions involving high-dimensional essential spheres as motivation; the question seeks further restrictions and their sharpness.

The requested conditions should be invariant under homeomorphism of the boundary. An obstruction that depends only on a particular drawing, cell subdivision, or selected visual metric does not by itself obstruct the topological boundary type.

Short English statement

What topological boundary types are ruled out by negative curvature together with hyperbolic cubical cell structure?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200008 (AMR-061-0008), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Kapovich - Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 8, PDF page 4. Original source indexed by AMR-061-0008. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.