Branched surfaces and triangulations — Question 7.7

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 branched surface $B\subset M$ is a smoothly tangent $2$-complex with the standard local branching models. Its fibered neighborhood $N(B)$ is an interval bundle with the usual identifications over the branch locus. A lamination is carried by $B$ if it lies in $N(B)$ transverse to interval fibers, and fully carried if it meets every fiber. Splitting means cutting this fibered neighborhood along carried surface pieces and taking the induced branched surface. A taut local orientation orders the vertices of each tetrahedron so that the star of each vertex is compatible with a local normal foliation and every directed loop in the oriented $1$-skeleton is homotopically essential. An amenable cover means a cover whose coset action of $\pi_1(M)$ on its fibers admits an invariant mean; this includes regular covers with amenable deck group.

Statement recorded in the supplied source.

If $B\subset M$ is the branched surface dual to a taut local orientation, must there be a finite covering $p:\widehat M\to M$ such that $p^{-1}(B)$ fully carries a lamination? Ask separately for an amenable covering. The branched surface is prescribed in advance and must be pulled back; existence of some unrelated lamination in a cover does not settle the question. [S2]

Short English statement

Does the pullback of a specified branched surface dual to a taut local orientation carry a lamination in a finite cover, or in an amenable cover?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10300021 (AMR-102-0021). 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 7.7; Definitions 1.1--1.10 and local definitions and remarks. https://arxiv.org/abs/math/0209081v1.