Boundaries of Groups and Kleinian Groups — Problem 28

Definitions and mathematical statement

A geodesic metric space $X$ is CAT(0) when every geodesic triangle satisfies Euclidean upper comparison. It is proper if its closed bounded sets are compact, and cocompact if its full isometry group has compact quotient. Its visual boundary $\partial_\infty X$ is the space of bounded-distance classes of geodesic rays, topologized by convergence on bounded intervals of representatives from a fixed basepoint.

Classify the nonempty compact metrizable spaces $Z$ for which there exists an unbounded proper cocompact CAT(0) space $X$ with \[ Z\cong\partial_\infty X, \] or find further effective necessary conditions for this realization. The source lists finite covering dimension, restrictions on the number of connected components, and nonzero top-dimensional Čech cohomology among the known conditions motivating the question.

There is a related, more restrictive realization problem in which a discrete group must act freely and cocompactly by isometries on $X$. In that setting the source also records dimensional homogeneity of the boundary: every nonempty open subset has the same covering dimension as the whole boundary. These two realization settings should be kept distinct; cocompactness of the full isometry group does not by itself supply a free discrete cocompact action.

The desired characterization concerns topology, not a particular boundary metric. The empty boundary of a bounded space is set aside explicitly so that the question addresses geometry at infinity.

Short English statement

Which compact spaces can occur as visual boundaries of proper cocompact CAT(0) spaces?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200028 (AMR-061-0028), 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 28, PDF page 9. Original source indexed by AMR-061-0028. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.