Boundaries of Groups and Kleinian Groups — Problem 9

Definitions and mathematical statement

A Coxeter system $(W,S)$ consists of a group with a finite generating set and presentation \[ W=\langle S\mid s^2=1,\ (st)^{m_{st}}=1 \text{ when }m_{st}<\infty\rangle, \] where $m_{st}=m_{ts}\in\{2,3,\ldots,\infty\}$ for distinct generators. For $T\subseteq S$, the special subgroup $W_T=\langle T\rangle$ is spherical if it is finite. The Davis complex $\Sigma(W,S)$ is built from cosets of these spherical subgroups, or equivalently their associated Coxeter cells, with its standard piecewise Euclidean CAT(0) metric.

The visual boundary consists of geodesic rays modulo bounded-distance equivalence, with the topology of convergence on compact intervals after rays are based at a common point. The group acts geometrically on its Davis complex, meaning properly and cocompactly by isometries.

If two finitely generated Coxeter groups are isomorphic as abstract groups, must the boundaries of their Davis complexes be homeomorphic? Equivalently, for two finite Coxeter generating sets $S,S'$ of the same group, must \[ \partial_\infty\Sigma(W,S)\cong\partial_\infty\Sigma(W,S')? \] The requested homeomorphism need not be equivariant. This is the source's question about boundaries furnished by Coxeter systems. It does not assert uniqueness for every CAT(0) space admitting an action of the group; arbitrary geometric actions form a broader class than the specified Davis constructions.

Short English statement

Does the abstract Coxeter group determine the homeomorphism type of its Davis-complex boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200009 (AMR-061-0009), 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 9, PDF page 4. Original source indexed by AMR-061-0009. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.