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是（1）将相关结构分解为它们的基本构建块和（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，可以研究对象纤维（即，可以由以可预测的方式布置的更简单、更低维度的部件构建）。即使对物体的完全理解可能遥不可及，人们也可以试图理解它的“平均”属性。“在这个项目中，PI试图通过它们对自然关联空间的作用，将类似的原则应用于几何重要群体的研究。这个追求带来承担的方法从几何学，拓扑学，动力学，和群论，并寻求不仅在抽象意义上理解这些对象，但产生易于处理的模型，允许定量的理解。更详细地说，PI将开展项目，调查的几何和拓扑双曲丛，以及研究几何动机计数问题在双曲生成群。在经典的设置双曲3-流形在圆上，这个项目建立在一个程序来研究转向三角剖分（某种组合构造）有效地编码流形的几何和拓扑的程度。受这些三角剖分的优雅性质的启发，PI将在自由循环群的设置中构建一个规范的理想单纯复形;一个当前缺乏控制群代数分解的单个拓扑对象的设置。最后，PI将研究纤维流形的“典型”几何，自由循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.