Boundaries of Groups and Kleinian Groups — Problem 10

Definitions and mathematical statement

A Coxeter group has a presentation by generators $s$ with $s^2=1$ and relations $(st)^{m_{st}}=1$ for finite Coxeter labels $m_{st}$; the label $\infty$ imposes no such relation. The source asks for a Coxeter group $G_n$ with boundary satisfying \[ \dim(\partial G_n)=n,\qquad \operatorname{hdim}_{{\mathbb{Q}}}(\partial G_n)=1. \] Here the first dimension is naturally interpreted as topological covering dimension: a compact space has dimension at most $n$ if every finite open cover has an open refinement of multiplicity at most $n+1$. Equality requires this and failure of the corresponding bound $n-1$.

The second symbol retains the source's phrase "rational homological dimension", whose definition is not given. A dimension defined using rational local or relative homology, rational cohomological dimension of compacta, and the largest degree with nonzero global rational homology are different notions. In particular, vanishing of high-degree global homology alone cannot silently stand for the requested dimension-one condition.

The boundary is also unspecified. A visual boundary of a chosen proper CAT(0) Coxeter complex, such as the Davis complex, and the Gromov boundary of a word-hyperbolic Coxeter group require different ambient hypotheses. The source does not require hyperbolicity or identify a chosen complex. Finally, $n$ has no explicit range or quantifier: existence for a prescribed $n$, for every $n$, and for arbitrarily large $n$ are distinct targets. The displayed conditions preserve the source's requested dimension gap but require these definitions and quantifiers before yielding a self-contained theorem.

Short English statement

Can a Coxeter-group boundary have topological dimension $n$ but rational homological dimension only one? The boundary model, rational dimension invariant, and quantifier on $n$ need to be identified.

Sources

• Local UnsolvedMath registry, permanent ID 6200010, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 10"; proposed by Alexander Dranishnikov, 2005. Registry status open, classification OPEN-TRIAGE.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 10, PDF page 4, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. No PDF was accessed here.
• The supplied summary mentions related Coxeter-boundary constructions with low homological versus topological dimension but supplies no precise construction, dimension convention, or publication. It says the stated case was not resolved in its accessible literature check; this is a supplied open/triage assessment.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the source is publicly author-hosted.