Boundaries of Groups and Kleinian Groups — Problem 20

Definitions and mathematical statement

For a proper CAT(0) space $X$, let $\partial_\infty X$ be the compact space of geodesic rays modulo bounded distance, with the cone topology. A group acts geometrically on $X$ when its action is isometric, proper and cocompact. Different proper CAT(0) spaces carrying geometric actions of the same group can have nonhomeomorphic visual boundaries.

Find useful topological invariants that distinguish such boundaries, and determine which boundary properties remain invariant under quasi-isometry of the underlying spaces. A map $f:X\to Y$ is a quasi-isometry if there are $L\geq1,C\geq0$ such that \[ L^{-1}d_X(x,x')-C\leq d_Y(fx,fx')\leq Ld_X(x,x')+C \] and every point of $Y$ is within distance $C$ of $f(X)$.

The source already records shape equivalence of boundaries for a fixed CAT(0) group and invariance of topological dimension in the proper cocompact setting. Consequently, merely recovering these invariants does not distinguish the known nonhomeomorphic examples. Shape equivalence preserves their Čech cohomology while allowing finer local topological differences.

The research problem asks both for invariants finer than these known ones that detect differences, and for coarser classifications preserved by large-scale geometry. A proposed answer should specify whether it applies to all proper cocompact CAT(0) spaces or only to boundaries associated with a fixed geometric group action.

Short English statement

Which boundary invariants detect differences between CAT(0) models, and which survive quasi-isometry?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200020 (AMR-061-0020), 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 20, PDF page 7. Original source indexed by AMR-061-0020. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.