Boundaries of Groups and Kleinian Groups — Problem 12

Definitions and mathematical statement

An integral homology $n$-manifold is a locally compact finite-dimensional space $Z$ such that for every $z\in Z$ its local singular homology satisfies \[ H_i(Z,Z\setminus\{z\};\mathbb Z)\cong \begin{cases}\mathbb Z&i=n,\\0&i\ne n.\end{cases} \] In the exotic examples motivating this question, $Z$ is also an Euclidean neighborhood retract: it embeds in a Euclidean space as a retract of an open neighborhood. A resolution is a surjective cell-like map $M^n\to Z$ from a topological $n$-manifold. Cell-like means that every point preimage has the shape of a point, equivalently is contractible inside each of its neighborhoods in an ambient absolute neighborhood retract.

Can an exotic, nonresolvable homology manifold of this kind occur as the visual boundary of a finitely generated Coxeter group? More precisely, does there exist a Coxeter system $(W,S)$ whose standard CAT(0) Davis complex has \[ \partial_\infty\Sigma(W,S)\cong Z \] for such a compact nonresolvable homology manifold?

The boundary uses bounded-distance classes of geodesic rays and the cone topology. A homology manifold need not be a topological manifold; local homology alone does not provide coordinate charts. The adjective exotic refers to the nonresolvable homology manifolds cited in the source, not merely to an unusual smooth structure on an ordinary manifold. Exhibiting a manifold boundary therefore would not answer this question.

Short English statement

Can a Coxeter-group boundary be an exotic homology manifold that has no manifold resolution?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200012 (AMR-061-0012), 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 12, PDF page 5. Original source indexed by AMR-061-0012. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.