Questions in Geometric Group Theory — Q 2.6

Definitions and mathematical statement

Let a discrete group $G$ act geometrically on two proper CAT$(0)$ geodesic spaces $X$ and $Y$. Here proper means that closed bounded balls are compact, and geometrically means by isometries, properly discontinuously, and with compact quotient. The CAT$(0)$ condition compares geodesic triangles with Euclidean triangles, requiring distances between corresponding side points not to increase.

The visual boundary $\partial X$ consists of geodesic rays modulo finite Hausdorff distance. Based at a point of $X$, each class has a unique representative, and uniform convergence on bounded time intervals defines the cone topology. This is a compact metrizable boundary; define $\partial Y$ similarly.

A compact metric space $C$ is cell-like if, after embedding it in the Hilbert cube, its inclusion into every neighborhood is null-homotopic in that neighborhood. This is the shape-of-a-point condition, independent of the embedding. A cell-like map is a continuous surjection whose point preimages are cell-like compacta.

Must there be a compact metric space $Z$ and cell-like maps \[ Z\longrightarrow\partial X,\qquad Z\longrightarrow\partial Y? \] No action of $G$ on $Z$, and no equivariance of these maps, is required. The question asks for a common cell-like resolution, even when the two boundaries are not homeomorphic. Properness of the spaces makes explicit the usual compact visual-boundary setting; it is not a request to identify the spaces themselves by an isometry.

Short English statement

Do two geometric CAT(0) actions of the same group have visual boundaries admitting a common cell-like resolution?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100206 (AMR-010-0206), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 2.6 (PDF page 7). Original source indexed by AMR-010-0206. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.