Boundaries of Groups and Kleinian Groups — Problem 11

Definitions and mathematical statement

Let $(W,S)$ be a Coxeter system with finite $S$ and symmetric Coxeter matrix $(m_{st})$, so $s^2=1$ and $(st)^{m_{st}}=1$ whenever $m_{st}<\infty$. Its Coxeter diagram records these exponents. Let $\Sigma(W,S)$ be the standard CAT(0) Davis complex, assembled from cells associated with finite special subgroups $\langle T\rangle$, $T\subseteq S$. Write $Z=\partial_\infty\Sigma(W,S)$ for the visual boundary, topologized by convergence of geodesic rays from a fixed basepoint.

For each integer $k\geq0$, find conditions on the Coxeter diagram which characterize, or give useful sharp criteria for, the following properties: $Z$ is $k$-connected, and $Z$ is locally $k$-connected.

Here $k$-connected means path-connected with $\pi_j(Z,z)=0$ for $1\leq j\leq k$. Local $k$-connectedness means that every neighborhood $V$ of a point contains a smaller neighborhood $U$ such that, for $0\leq j\leq k$, every map \[ S^j\longrightarrow U \] extends to a map $B^{j+1}\to V$. In degree zero this includes connecting pairs of nearby points by paths in the prescribed larger neighborhood.

These are separate global and local requirements. Neither is replaced by vanishing of homology alone. A criterion should state which integer $k$ it controls and should apply to the visual boundary, rather than only to the finite nerve or the contractible Davis complex itself.

Short English statement

Which conditions on a Coxeter diagram determine global and local connectivity of the visual boundary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200011 (AMR-061-0011), 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 11, PDF page 4. Original source indexed by AMR-061-0011. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.