Boundaries of Groups and Kleinian Groups — Problem 101

Definitions and mathematical statement

Let $\Lambda\subset\partial_\infty\mathbb H^3$ and let $C(\Lambda)$ denote its hyperbolic convex hull. For a point $p$ in that hull, write \[ d(p)=\operatorname{dist}_{\mathbb H^3}\left(p,\mathbb H^3\setminus C(\Lambda)\right). \] This measures depth inside the hull; for a point outside the hull the distance to the exterior is zero. A quasicircle is the image of a round circle under a quasiconformal self-homeomorphism of the boundary sphere.

The source asks for an estimate of the local bi-Lipschitz constant of a map denoted $Q_\Lambda$ near $p$, in terms of $d(p)$, and asks specifically whether the decay is exponential when $\Lambda$ is a quasicircle. The map $Q_\Lambda$, its domain, codomain, and metrics are not defined in the supplied record.

For a specified map between metric spaces, a local $L$-bi-Lipschitz estimate would require a neighborhood $U$ of $p$ such that \[ L^{-1}d_X(x,y)\leq d_Y(Q_\Lambda x,Q_\Lambda y)\leq Ld_X(x,y) \qquad(x,y\in U),\qquad L\geq1. \] The neighborhood and whether an infinitesimal optimal constant is intended are unspecified. Since $L\geq1$, literal exponential decay of $L$ to zero cannot be the intended asymptotic. Possible precise targets involve $L-1$ or $\log L$, for example $L(p)-1\leq Ce^{-c d(p)}$, but the source does not select such an expression or state how $C,c$ may depend on $\Lambda$ or its quasiconformal distortion. No particular estimate is asserted without these definitions.

Short English statement

Does a specified but unidentified map associated with a quasicircle become exponentially close to locally isometric deep inside its convex hull? The map, distortion normalization, and uniformity of constants are missing.

Sources

• Local UnsolvedMath registry, permanent ID 6200101, exact source code catalogue entry, "Boundaries of Groups and Kleinian Groups---Problem 101"; proposed year 2005, no proposer supplied. Registry status open, classification OPEN-TRIAGE.
• Kapovich, Problems on Boundaries of Groups and Kleinian Groups (2005), Problem 101, PDF page 25, https://www.math.ucdavis.edu/~kapovich/EPR/problems.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026. The definition of $Q_\Lambda$ is absent; no PDF was accessed here.
• The supplied research summary relates this to quasiconformal boundary maps, quasisymmetric extension, convex hulls and quasicircles, without a precise defining reference, and reports no verified estimate. This is supplied open/triage assessment.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the source is publicly author-hosted.