Veering triangulations and Cannon-Thurston maps (Q2826653)

For a hyperbolic surface bundle over the circle \(M\) and a fiber surface \(\Sigma\), the inclusion map \(\Sigma\to M\) induces a map on the universal cover \(\widetilde{\Sigma}=\mathbb{H}^2\to \widetilde{M}=\mathbb{H}^3\). It is known that this map extends to a surjective map on the boundary \(S^1_{\infty}=\partial_{\infty}\mathbb{H}^2\to S^2_{\infty}=\partial_{\infty}\mathbb{H}^3\), which is called the Cannon-Thurston map.NEWLINENEWLINEFor a hyperbolic surface bundle \(M\) with all singularities of the invariant foliation on \(\Sigma\) occuring at the punctures, Agol constructed an ideal triangulation of \(M\). This triangulation is naturally associated to the suspended foliation, and it is called the veering triangulation.NEWLINENEWLINEFor such hyperbolic surface bundles, the author gives a combinatorial description of the Cannon-Thurston map, in terms of the veering triangulation. Between the Cannon-Thurston map and veering triangulation, the Euclidean structure on \(\widetilde{\Sigma}\) shows up naturally as a bridge.NEWLINENEWLINEMore precisely, starting with the Euclidean structure on \(\widetilde{\Sigma}\), the author gives an alternative construction of the veering triangulation of the \(3\)-manifold. On the other hand, after fixing a singularity \(\underline{\Omega}\) of \(\widetilde{\Sigma}\), the author gives a combinatorial description of the Cannon-Thurston map, with coordinates such that the cusp corresponding to \(\underline{\Omega}\) is identified with \(\infty\in S^2_{\infty}=\mathbb{C}\cup\{\infty\}\).NEWLINENEWLINEWith these two connections, the author gives a relation between the link of the \(\underline{\Omega}\)-cusp in the veering triangulation and the combinatorial structure of the Cannon-Thurston map, with the \(\underline{\Omega}\)-cusp identified with \(\infty \in S^2_{\infty}\). Basically these two combinatorial structures determine each other.NEWLINENEWLINEIn particular, the combinatorial structure of the Cannon--Thurston map is very interesting, and it is described by the following data:NEWLINENEWLINE1. A \(\mathbb{Z}\)-family of Jordan curves \(J_i\) in \(\mathbb{C}\cup\{\infty\}\) going through \(\infty\), such that \(J_i\cap J_{i'}=\{\infty\}\) if and only if \(|i-i'|>1\).NEWLINENEWLINE2. Each \(J_i\) bounds a domain \(D_i\), such that \(\cdots\supset D_{-1}\supset D_0\supset D_1\supset \cdots\), with \(\cap D_i=\emptyset\) and \(\cup D_i=\mathbb{C}\).NEWLINENEWLINE3. For each \(i\), the closure of \(D_i\setminus D_{i+1}\) in \(\mathbb{C}\) is the union of a \(\mathbb{Z}\)-family of closed disks \(\{\delta_s^i\}_{s\in \mathbb{Z}}\), such that they are all disjoint except \(\delta_s^i\cap \delta_{s+1}^i\) is one point.NEWLINENEWLINE4. The Cannon-Thurston map fills out \(D_i\setminus D_{i+1}\) by filling \(\delta_s^i\) one by one, and the order of the filling depends on the parity of \(i\).NEWLINENEWLINEThe author also gives many illustrations of the link of veering triangulations and the combinatorial structure of Cannon-Thurston maps at the end of this paper.