Boundaries of Groups and Kleinian Groups — Problem 71

Definitions and mathematical statement

Let $G$ be a finitely generated group. The source proposes a compactification constructed in a Stone--Čech manner from first $\ell^2$ cohomology, or from another function-space cohomology, and asks whether its associated incidence geometry at infinity is always "trivial (i.e., hyperbolic)".

One standard model for first $\ell^2$ cohomology uses the left regular representation $\lambda$ on $\ell^2(G)$: cocycles satisfy $b(gh)=b(g)+\lambda(g)b(h)$ and coboundaries have the form $b(g)=\lambda(g)v-v$. Reduced cohomology further closes the coboundaries. The source does not choose reduced versus unreduced cohomology or specify a model of representing functions.

For a family $\mathcal F$ of bounded scalar functions, a function compactification can be described by the closure of the evaluation image \[ G\longrightarrow\prod_{\varphi\in\mathcal F}\overline{\varphi(G)},\qquad g\longmapsto(\varphi(g))_{\varphi\in\mathcal F}, \] provided the necessary separation and embedding conventions hold. But a cohomology space is not automatically such an algebra or separating family of bounded functions. The record does not identify $\mathcal F$, the treatment of representatives and unbounded functions, or a translation-invariant closure.

It also supplies no incidence relation, objects at infinity, or definition of triviality. In particular, the parenthetical word hyperbolic must not be turned into a claim that $G$ is word-hyperbolic without an explanation of that equivalence. Thus the compactification and incidence construction are essential missing definitions, and the source determines a research question rather than a unique quantified theorem.

Short English statement

For a group compactification built from first $\ell^2$ or related function-space cohomology, must its geometry at infinity be trivial in the source's hyperbolic sense? The compactification and incidence geometry are not defined.

Sources

• Local UnsolvedMath registry, permanent ID 6200071, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 71"; proposed by Anders Karlsson, 2005. Registry status open, classification OPEN-TRIAGE.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 71, PDF page 19, 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 research summary names Bestvina and Gromov for related $\ell^2$-cohomology/boundary work without precise references and calls the incidence-geometry interpretation open. It does not define the objects in the question or establish a current resolution.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the source is publicly author-hosted.