Subsurface distances for hyperbolic 3-manifolds fibering over the circle (Q6594555)

Let \(M\) be a hyperbolic fibered 3-manifold. If \(S\) is a fiber in a fibration of \(M\), then the stable and unstable laminations \(\lambda^{\pm}\) on \(S\) are induced by the corresponding pseudo-Anosov homeomorphism on \(S\). The veering triangulation is defined as an ideal triangulation of \(\mathring{M}\), where \(\mathring{M} \subset M\) is the exterior of the singular points of the pseudo-Anosov homeomorphism. The 3-manifold \(M\) admits many fibrations in general. The Thurston norm on \(H^{1}(M; \mathbb{R})\) has a polyhedral unit ball with specified top-dimensional open faces, called the fibered faces. Each fiber of \(M\) is dual to a cohomology class in the cone over a fibered face. The veering triangulation depends only on a fibered face. For a subsurface \(Y\) of a fiber \(S\), the subsurface projection distance \(d_{Y}(\lambda^{-}, \lambda^{+})\) is defined through the curve graph, and measures the complexity of \(\lambda^{\pm}\) as seen from \(Y\).\N\NThe main theorems in this paper concern bounds of \(d_{Y}(\lambda^{-}, \lambda^{+})\) as follows. First, for any fiber \(S\) contained in the cone over a fibered face \(\mathbf{F}\) and any subsurface \(Y\) of \(S\), it holds that\N\[\N\max\{|\chi(Y)|, 1\} \cdot (d_{Y}(\lambda^{-}, \lambda^{+}) - 240) \leq 30 |\mathbf{F}|,\N\]\Nwhere \(|\mathbf{F}|\) is the number of tetrahedra of the veering triangulation associated to \(\mathbf{F}\). Second, for fibers \(S\) and \(F\) in the same fibered cone and a subsurface \(W\) of \(F\), either\N\begin{itemize}\N\item \(W\) is homotopic through surfaces transverse to the associated flow to an embedded subsurface \(W'\) of \(S\) with \(d_{W'}(\lambda^{-}, \lambda^{+}) = d_{W}(\lambda^{-}, \lambda^{+})\), or\N\item \(135 |\chi(S)| \geq d_{W}(\lambda^{-}, \lambda^{+}) - 240\).\N\end{itemize}\NFurthermore, the authors show that if \(Y \to S\) is an immersion to a finitely generated subgroup of \(\pi_{1}(S)\), then either\N\begin{itemize}\N\item there is a subsurface \(W \subset S\) so that \(Y \to S\) through a finite cover \(Y \to W\), or\N\item the diameter of image of the projection \(\mathcal{A}(S) \to \mathcal{A}(Y)\) is at most 38.\N\end{itemize}\N\NThe first and second results are extensions of a previous work of the authors [Geom. Funct. Anal. 27, No. 6, 1450--1496 (2017; Zbl 1385.57025)] in the case that \(\mathring{M} = M\). To show them, the authors develop tools on combinatorics of veering triangulations and subsurface projections.