Boundaries of Groups and Kleinian Groups — Problem 21

Definitions and mathematical statement

Let a group $G$ act geometrically on two proper CAT(0) spaces $X_1,X_2$. Thus both actions are proper, cocompact and isometric, and each space satisfies Euclidean upper triangle comparison. Let $Z_i=\partial_\infty X_i$ be its visual boundary, defined by bounded-distance classes of geodesic rays with the cone topology.

A compact metric space $K$ is cell-like if it has the shape of a point. Concretely, after embedding it in a Hilbert cube, it can be contracted to a point inside every neighborhood of its image. A continuous surjection is cell-like when every point preimage is cell-like.

Must there exist a compact metrizable space $Z$ and continuous surjections \[ Z\xrightarrow{\ p_1\ }Z_1,\qquad Z\xrightarrow{\ p_2\ }Z_2 \] such that both $p_1$ and $p_2$ are cell-like? This is the common-domain formulation explicitly requested in the source.

Neither $Z$ nor the two maps are required to carry compatible $G$-actions. Requiring equivariance gives a stronger problem. The assertion also does not require either map to be injective: collapsing cell-like fibers is what permits different homeomorphism types. Ordinary shape equivalence is already known in this setting, but a statement of shape equivalence alone does not construct the specific common compactum and cell-like maps requested here. The same group, rather than two unrelated groups, acts on both CAT(0) models.

Short English statement

Do any two CAT(0) boundaries of the same group have a common compactum mapping cell-like onto both?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200021 (AMR-061-0021), 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 21, PDF page 7. Original source indexed by AMR-061-0021. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.