Boundaries of Groups and Kleinian Groups — Problem 37

Definitions and mathematical statement

The source asks for useful classes of spaces on which an infinitesimal metric condition for quasiconformality implies a local condition, and notes Loewner spaces as a known setting. The two conditions themselves are not reproduced in the excerpt.

For orientation, for a homeomorphism $f:(X,d)\to(Y,\rho)$, a common infinitesimal metric quantity is \[ H_f(x)=\limsup_{r\downarrow0}\frac{L_f(x,r)}{l_f(x,r)}, \] where \[ L_f(x,r)=\sup_{d(x,a)\le r}\rho(f(x),f(a)),\qquad l_f(x,r)=\inf_{d(x,b)\ge r}\rho(f(x),f(b)), \] for radii and spaces where these quantities are meaningful. An infinitesimal bounded-distortion condition may require $H_f(x)\le H$ uniformly; a possible local comparison condition is local weak quasisymmetry, controlling \[ d(x,a)\le d(x,b)\quad\Longrightarrow\quad \rho(f(x),f(a))\le C\rho(f(x),f(b)) \] for triples in suitable neighborhoods. Full local quasisymmetry gives a control function for arbitrary ratios instead. These are illustrative conventional notions, not an identification of the missing source definitions. Whether the infinitesimal requirement is everywhere or almost everywhere, and how the local constants are uniform, also matters.

With the intended predicates fixed, the task is to identify classes of domain/codomain spaces for which every eligible homeomorphism satisfying the infinitesimal predicate satisfies the local predicate, with the intended quantitative dependence. The source gives no exact maximality criterion for "good classes".

The usual Loewner framework involves a metric measure structure and quantitative modulus bounds for curve families joining continua; dimension and regularity assumptions are material to the relevant equivalence theorems. The supplied summary attributes a developed theory to Heinonen--Koskela and calls the record survey material, but gives no exact theorem hypotheses for domain, target, or measure. The registry nevertheless labels it open. This can preserve a known special setting together with an open-ended search for further classes; it does not establish a universal implication for unspecified spaces.

Short English statement

Identify geometric classes where infinitesimal metric quasiconformality forces the intended local distortion control. Loewner spaces are cited as a known setting, but the excerpt omits the precise predicates and hypotheses needed to state an implication theorem.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6200037$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Mario Bonk (2005), Problem 37 in Kapovich, Problems on Boundaries of Groups and Kleinian Groups, PDF page 12: https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized PDF-text extraction accessed 2026-07-29; current status marked as needing review.
• Heinonen--Koskela Loewner-space and quasiconformal theory is named by the summary without a full publication citation or exact equivalence theorem.