On the combinatorics of veering triangulations

在1970年的《S和1980年的S》中，威廉·瑟斯顿彻底改变了低维拓扑学。他向这个领域引入了许多新的思想，并进一步揭示了三维流形的组合研究与许多其他领域之间令人兴奋的联系：TeichMuller理论、全纯动力学、克莱因群、几何群论等。他开创的新技术之一是使用理想三角剖分来表示双曲三维流形。这使得拓扑学家一方面可以使用代数几何中的工具，另一方面可以为大杂烩般的不同例子提供组织原则。理想三角剖分的概念现在已经被推广和特化了许多次，得到了有角度的三角剖分[Casson]，紧理想三角剖分[Lackenby]和转向三角剖分[Agol，HRST]。Francois Gueritaud发现了纤维流形的Cannon-瑟斯顿映射和它们的转向三角剖分之间的重要联系。Yair Minsky和Samuel Taylor给出了地下投影机制和转向三角剖分结构之间的有趣关系。这个项目的目标是了解在非纤维环境下转向三角剖分的组合如何告知响应的坎农-瑟斯顿地图和三个流形的万能盖子的纤维结构。

In the 1970's and 1980's William Thurston revolutionised low-dimensional topology. He introduced many new ideas into the field, and furthermore revealed exciting connections between the combinatorial study of three-manifolds and many other areas:Teichmuller theory, holomorphic dynamics, Kleinian groups, geometric group theory, and more.One of the new techniques he pioneered was the use of ideal triangulations to represent hyperbolic three-manifolds. This allowed topologists, on the one hand, access to tools in algebraic geometry and, on the other hand, an organising principle for what had been a hodge-podge of disparate examples. The idea of ideal triangulations has now been generalised and specialised many times to give angled triangulations [Casson], taut ideal triangulations [Lackenby], and veering triangulations [Agol, HRST].Francois Gueritaud has found an important connection between the Cannon-Thurston map for fibered manifolds and their veering triangulations. Yair Minsky and Samuel Taylor have given an interesting relation between the machinary of subsurface projections and the structure of veering triangulations. The goal of this project is to understand how the combinatorics of a veering triangulation, in the non-fibered setting, inform the cooresponding Cannon-Thurston map and the "fibered structure" of the universal cover of the three-manifold.