Negative Curvature in Fiber Bundles and Counting Problems

在许多可用于探索复杂几何系统的技术中，通常最有成效的两种技术是：(1)将相关结构分解为其基本构建块；(2)从这些系统中采样以确定其典型行为。例如，为了理解高维几何物体，可以研究物体纤维的各种方式（即，可以用更简单的、以可预测的方式排列的低维碎片构建）。即使完全理解一个对象可能是遥不可及的，一个人可以尝试理解它的性质“平均”。在这个项目中，PI试图通过对自然相关空间的作用，将类似的原理应用于几何上重要群体的研究。这种追求带来了几何学、拓扑学、动力学和群论的方法，并寻求不仅在抽象意义上理解这些对象，而且产生可处理的模型，允许定量理解。更详细地说，PI将开展研究双曲束的几何和拓扑的项目，以及研究有限生成群中的几何驱动计数问题。在圆上的双曲3流形光纤的经典设置中，本项目建立在一个程序的基础上，研究转向三角测量（某种组合结构）有效编码流形几何和拓扑的程度。受到这些三角形的优雅性质的启发，PI将在自由环群的设置中构建一个规范的理想简单复合体；目前还缺乏单个拓扑对象控制群的代数分解的设置。最后，PI将研究纤维流形的“典型”几何，自由环群和双曲群的表示。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.