Boundaries of Groups and Kleinian Groups — Problem 17

Definitions and mathematical statement

A group $G$ is a CAT(0) group if it acts properly and cocompactly by isometries on a proper geodesic CAT(0) space. CAT(0) means geodesic triangles are no thicker than their Euclidean comparison triangles. The visual boundary $\partial_\infty X$ of such a space consists of geodesic rays modulo bounded distance, with the topology of uniform convergence on bounded intervals for representatives from a common basepoint.

Suppose $G$ is one-ended, so deleting any finite vertex set from a Cayley graph leaves exactly one infinite component. Suppose $G$ acts geometrically on $X$ and $\partial_\infty X$ is locally connected: every point has a basis of connected open neighborhoods. Must every other proper CAT(0) space $Y$ carrying a geometric action of $G$ satisfy \[ \partial_\infty X\cong\partial_\infty Y? \] The homeomorphism sought is topological and need not be $G$-equivariant. One may also isolate the weaker subquestion in which both boundaries are assumed locally connected.

The same group acts on both spaces, but its actions and their metrics may be different. The question does not require the quasi-isometry between the spaces supplied by their orbit maps to extend continuously to the boundaries. Thus failure of a particular boundary extension is not by itself a counterexample: a counterexample must rule out every homeomorphism between the boundary spaces.

Short English statement

Does having one locally connected CAT(0) boundary force a one-ended group’s visual boundary to be unique?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200017 (AMR-061-0017), 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 17, PDF page 6. Original source indexed by AMR-061-0017. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.