Boundaries of Groups and Kleinian Groups — Problem 38

Definitions and mathematical statement

Let $(Z,d,\mu)$ be a compact metric measure space. It is Ahlfors $Q$-regular if there is $C\geq1$ such that \[ C^{-1}r^Q\leq\mu(B(z,r))\leq Cr^Q \qquad(0<r\leq\operatorname{diam}Z). \] For a family $\Gamma$ of rectifiable curves, its $Q$-modulus is the infimum of $\int_Z\rho^Q\,d\mu$ over nonnegative Borel functions $\rho$ satisfying $\int_\gamma\rho\,ds\geq1$ for every $\gamma\in\Gamma$.

For disjoint nondegenerate continua $E,F\subset Z$, let $\Gamma(E,F)$ be the curves joining them and put $\Delta(E,F)=d(E,F)/\min(\operatorname{diam}E,\operatorname{diam}F)$. The $Q$-Loewner property requires a positive decreasing function $\phi$ such that \[ \operatorname{Mod}_Q\Gamma(E,F)\geq\phi(\Delta(E,F)) \] for every such pair. This is a lower modulus bound; reversing its direction would not define the Loewner condition.

Identify and characterize boundaries of hyperbolic groups or hyperbolic spaces, beyond the Fuchsian-building families emphasized in the source, that have this property for an Ahlfors-regular metric in their visual quasisymmetry class. Visual metrics measure how long boundary rays fellow-travel; quasisymmetric changes control ratios of distances by a fixed distortion function.

The metric and measure must be specified with the boundary, since the condition is analytic and not just a property of its homeomorphism type. The problem seeks new families and useful criteria, not merely a restatement that the known examples satisfy the defining inequality.

Short English statement

Which group boundaries, beyond the classical Fuchsian-building examples, support Loewner analysis?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 6200038 (AMR-061-0038), 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 38, PDF page 12. Original source indexed by AMR-061-0038. https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf.