Hyperbolic geometry — Question 10.3

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A codimension-one foliation $\mathcal F$ decomposes a $3$-manifold $M$ into surface leaves, locally as the horizontal slices of a product chart. In the terminology of the source, it is taut if a closed curve transverse to $\mathcal F$ meets every leaf. A minimal foliation means that every leaf is dense; a minimal surface means a surface with vanishing mean curvature for the specified Riemannian metric. These two uses of minimal are distinct. For a finite-depth foliation, depth-zero leaves are compact; a leaf has depth $d>0$ when its frontier is a union of leaves of smaller depth and $d$ is the smallest such finite bound. The depth of the foliation is the greatest leaf depth. For a knot, depth is the minimum depth of a taut foliation of its exterior with the standard boundary behavior used for knot-depth foliations; fibered knots have depth zero. For a lifted leaf $\lambda\subset\mathbb H^3$, its limit set $\lambda_\infty$ is the intersection of its closure in $\mathbb H^3\cup S^2_\infty$ with $S^2_\infty$. Hausdorff dimension uses the round sphere metric, or an equivalent visual metric in the usual hyperbolic normalization.

Statement recorded in the supplied source.

For a finite-depth foliation $\mathcal F$ of a hyperbolic $3$-manifold, determine whether, and how, $\dim_H(\lambda_\infty)$ for a lifted leaf $\lambda$ is related to the depth of its underlying leaf or to the depth of $\mathcal F$. Do not restrict to compact leaves or depth one. [S2]

Short English statement

Relate the dimension of a lifted leaf limit set to leaf depth and total foliation depth.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300041 (AMR-102-0041). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Danny Calegari, Problems in foliations and laminations of 3-manifolds, version 0.78 (2002), Question 10.3; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.