Kirby Problem 3.25

Definitions and mathematical statement

Let $S$ be an oriented finite-type surface with $\chi(S)<0$. A marked quasi-Fuchsian structure is a discrete faithful representation $\rho:\pi_1(S)\to\operatorname{PSL}_2(\mathbb C)$ obtained by a quasiconformal deformation of a Fuchsian representation, modulo conjugacy; puncture loops remain parabolic. Its manifold $M_\rho$ is homeomorphic to $S\times\mathbb R$. The convex core is the quotient of the convex hull of the limit set in $\partial_\infty\mathbb H^3$.

The two marked boundary components of this core have intrinsic complete hyperbolic metrics $m_+,m_-$. They are pleated along measured geodesic laminations $\lambda_+,\lambda_-$, where a lamination is a closed union of disjoint complete geodesics and its transverse measure records the exterior bending angle. Thus one obtains maps \[ \mu:\mathcal QF(S)\to\mathcal T(S)^2,\qquad \beta:\mathcal QF(S)\setminus\mathcal F(S)\to\mathcal{ML}(S)^2. \] Here $\mathcal T(S)$ is marked hyperbolic-metric space, $\mathcal{ML}(S)$ is measured-lamination space, and $\mathcal F(S)$ is the Fuchsian locus.

(a) Is $\mu$ injective? (b) When $S$ has punctures, is $\beta$ injective? Equality includes the markings and the ordering of the two ends. In (a) a Fuchsian core is read as two copies of its intrinsic surface metric. In (b) the Fuchsian locus is excluded because its bending measures vanish. The unpunctured bending-uniqueness case is not being reasserted as open.

Short English statement

Do the two intrinsic convex-core metrics determine a marked quasi-Fuchsian manifold, and do the two bending measures determine it in the punctured non-Fuchsian case?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2823 (KP-3.25), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.25. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.