Boundaries of Groups and Kleinian Groups — Problem 63

Definitions and mathematical statement

Let $G$ be a word-hyperbolic group and let $(X,d,\mu)$ be the intended metric-measure realization of $X=\partial_\infty G$. In a usual $Q$-Loewner setting, $Q>1$, the measure is Ahlfors $Q$-regular and the $Q$-modulus of curves joining disjoint nondegenerate continua has a positive lower bound depending only on their relative distance. The source does not fix $Q$, the metric, or the measure.

A nonnegative Borel function $g$ is an upper gradient of a real function $u$ if every rectifiable curve $\gamma$ joining $x$ to $y$ satisfies \[ |u(x)-u(y)|\le\int_\gamma g\,\mathrm ds. \] Under the usual weak $(1,1)$-Poincaré convention, supporting a $1$-Poincaré inequality means that constants $C>0$ and $\lambda\ge1$ exist so that, for every ball $B=B(x,r)$, admissible function $u$, and its upper gradient $g$, \[ \frac1{\mu(B)}\int_B|u-u_B|\,\mathrm d\mu \le Cr\,\frac1{\mu(\lambda B)}\int_{\lambda B}g\,\mathrm d\mu, \qquad u_B=\frac1{\mu(B)}\int_Bu\,\mathrm d\mu. \] One can initially test this with Lipschitz $u$; the full test class, weak-upper-gradient variant, and whether dilation is allowed require specification.

Heinonen's question is to determine assumptions on $G$, beyond having a $Q$-Loewner boundary, that guarantee the boundary supports the intended $1$-Poincaré inequality. The exponent $1$ concerns the upper-gradient integral; a Poincaré inequality with a different exponent does not settle this endpoint question. No particular extra group hypothesis or necessary-and-sufficient characterization is supplied in the source.

Short English statement

Which additional properties of a hyperbolic group with a $Q$-Loewner boundary force an endpoint $1$-Poincaré inequality on that boundary? Metric, measure, and the precise Poincaré convention must be fixed.

The supplied summary mentions Bourdon--Kleiner and Poincaré consequences of Loewner properties but gives no exact exponent, theorem, or reference. It calls the $p=1$ question subtle and open. The OPEN-TRIAGE label and 2005 source are retained with current status explicitly marked for review.

Sources

• Local UnsolvedMath registry, permanent ID 6200063, source code catalogue entry, "Boundaries of Groups and Kleinian Groups --- Problem 63," proposed by Juha Heinonen, 2005. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 63, PDF page 17, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf; normalized pdf-text extraction, accessed 2026-07-29.
• The supplied research summary mentions Bourdon--Kleiner without complete bibliographic details or a precise endpoint result. Source-specific provenance marks current status and statement-reuse terms NEEDS_REVIEW for the public author-hosted PDF, separately from the dataset licence.