Foliated Teichmüller theory — Question 11.3

Definitions and mathematical statement

Let $M$ be an atoroidal three-manifold and $\mathscr F$ a foliation arising from the source's slithering over $S^1$. Let $X$ be a pseudo-Anosov flow transverse to $\mathscr F$, and suppose its time-one map $Z$ preserves the foliation. Lift the data to $\widetilde M$. For a leaf $\lambda$ and its images $Z^n(\lambda)$, use the hyperbolic uniformizations by $\mathbb H^2$ supplied in the statement via Candel's theorem. Thus an iterate restricts to a map between two hyperbolic leaves, after choices of identifications.

The approximation question asks for integers $n_i$, compact hyperbolic surfaces $\Sigma_i$, and Teichmüller representatives $\phi_i:\Sigma_i\to\Sigma_i$ of their mapping classes, with compatible lifts/markings, such that \[ F_i=\widetilde{\phi_i^{-1}}\circ Z^{n_i}:\mathbb H^2\longrightarrow\mathbb H^2 \] is a $k_i$-quasi-isometry with $k_i\to1$. A Teichmüller representative refers to the extremal quasiconformal representative of an isotopy class, not an arbitrary homeomorphism or an isometry.

Several data needed for precision are absent. The source does not require $n_i$ positive, nonzero, or tending to infinity; permitting $n_i=0$ and identity mapping classes makes the displayed existence condition trivial. A nontrivial iterated-approximation requirement therefore needs an explicit restriction, not a silently inserted $n_i\to\infty$.

The one-constant quasi-isometry convention is also unspecified. Usually two constants appear in inequalities $K^{-1}d-A\leq d(Fx,Fy)\leq Kd+A$, together with coarse surjectivity. Requiring $K\to1$ alone, requiring additive error also to vanish, and a bi-Lipschitz bound are different assertions. The source does not select among them.

Finally, the slithering/flow conventions and global category of $M$, the surface boundary convention, and the identifications relating leaf uniformizations to universal covers of $\Sigma_i$ must be supplied. Compact closed hyperbolic surfaces have universal cover $\mathbb H^2$; compact surfaces with geodesic boundary require a different covering description. No stronger Teichmüller-distance or convergence conclusion is inferred from the printed composition.

Short English statement

Approximate iterated leaf maps of a circle-slithering pseudo-Anosov flow by lifts of compact-surface mapping classes with distortion tending to one. The iterate nontriviality, lift identifications and distortion convention are missing.

Sources

• Local UnsolvedMath registry, permanent ID 10300049, exact source code catalogue entry, "Foliated Teichmüller theory---Question 11.3"; no proposer or proposed year in metadata. Registry status open, classification OPEN-TRIAGE.
• Calegari, Problems in foliations and laminations of 3-manifolds (2002), Question 11.3, https://arxiv.org/abs/math/0209081. Supplied source-TeX extraction accessed July 29, 2026; the author list has 67 numbered questions.
• The statement invokes Candel uniformization without a separate full citation. The supplied summary describes the approximation as an open rigidity question and reports no verified result, but supplies no precise later reference or missing quasi-isometry convention.
• Provenance flags current primary-literature status and redistribution terms NEEDS_REVIEW.