Negative Curvature in Fiber Bundles and Counting Problems

在探索复杂几何系统的许多技术中，通常最有成效的两种技术是(1)将相关结构分解成它们的基本构件，(2)从这些系统中采样以确定它们的典型行为。例如，为了理解高维几何对象，人们可以研究对象纤维的各种方式(即，可以用以可预测的方式排列的更简单、更低维的碎片来构建)。即使对物体的完全理解可能是遥不可及的，人们也可以尝试“平均地”理解它的属性。在这个项目中，PI试图通过它们在自然相关空间上的作用，将类似的原理应用于几何意义上的群的研究。这种追求运用了几何学、拓扑学、动力学和群论的方法，不仅试图从抽象的意义上理解这些对象，而且试图产生允许定量理解的易于处理的模型。更详细地，PI将执行研究双曲丛的几何和拓扑以及研究有限生成群中的几何激励计数问题的项目。在圆周上双曲三维流形的经典设置中，这个项目建立在一个程序的基础上，以研究转向三角剖分(某种组合结构)在多大程度上有效地编码了流形的几何和拓扑。受这些三角剖分的优雅性质的启发，PI将在自由循环群的背景下构建规范的理想单纯复形；在这种背景下，目前缺乏控制群的代数分解的单个拓扑对象。最后，PI将学习纤维流形、自由循环群和双曲群表示的“典型”几何。贯穿这些研究的一个共同主线将是证明负曲率特征在多大程度上是持久的，并且在适当的意义上是通用的。

Among the many techniques available for exploring complicated geometric systems, often two of the most fruitful are (1) decomposing related structures into their basic building blocks and (2) sampling from those systems to determine their typical behavior. For example, to understand a high dimensional geometric object, one can study the various ways that object fibers (i.e., can be build out of simpler, lower dimensional pieces which are arranged in a predictable way). Even if a complete understanding of the object may be out of reach, one can attempt to comprehend its properties "on average." In this project, the PI seeks to apply similar principles to the study of geometrically significant groups via their action on naturally associated spaces. This pursuit brings to bear methods from geometry, topology, dynamics, and group theory, and seeks to not only understand these objects in an abstract sense, but to produce tractable models which allow for quantitative understanding.In more detail, the PI will carry out projects that investigate the geometry and topology of hyperbolic bundles as well as study geometrically motivated counting problems in finitely generated groups. In the classical setting of a hyperbolic 3-manifold fibering over the circle, this project builds on a program to study the extent to which the veering triangulation (a certain combinatorial construction) effectively encodes the manifold's geometry and topology. Inspired by the elegant nature of these triangulations, the PI will construct a canonical ideal simplicial complex in the setting of free-by-cyclic groups; a setting where a single topological object controlling the group's algebraic decompositions is currently lacking. Finally, the PI will study the "typical" geometry of fibered manifolds, free-by-cyclic groups, and representations of hyperbolic groups. A common thread throughout these investigations will be to demonstrate the extent to which negative curvature features are persistent and, in the appropriate sense, generic.