Asymptotic methods in groups and locally symmetric spaces

通过空间的几何极限来理解一类空间或一系列空间通常是有用的。PI提出了将这种渐近方法应用于拓扑，几何和群论的五个项目。首先，PI将研究闭双曲3-流形的大尺度几何，其基本群可以由固定数量的元素生成。其次，PI将研究凸紧柄体的代数和几何极限，以及它们与3流形边界同胚向其内部扩展的关系。在群论中，PI希望研究某些自由群自同构的映射环面的拟等距刚性和群的所有指标n子群的交集的指标的渐近增长。最后，PI希望促进和扩展李群的“不变随机子群”这一激动人心的新课题，他和albert等人利用这一课题来控制高阶局部对称空间的Betti数的增长。在过去的40年里，二维和三维几何形状的研究迅速发展，部分原因是由于威廉·瑟斯顿在20世纪70年代的革命性工作。瑟斯顿是最早意识到“几何退化”在这些低维空间中的特殊效用的人之一。几何退化可以被认为是形状的变形，最终可能改变其类型：例如，半径缩小的球体退化为点，高度固定且周长缩小的垂直圆柱体退化为线段。该项目的目标是使用几何退化来分析各种低维形状，特别是三维流形，我们的宇宙就是一个例子。

It is often useful to understand a class or sequence of spaces through its geometric limits. The PI proposes five projects applying such asymptotic methods to topology, geometry and group theory. First, the PI will study the large-scale geometry of closed hyperbolic 3-manifolds whose fundamental groups can be generated by a fixed number of elements. Second, the PI will study algebraic and geometric limits of convex cocompact handlebodies and their relation to the extension of homeomorphisms of a 3-manifold's boundary into its interior. In group theory, the PI hopes to investigate the quasi-isometric rigidity of mapping tori of certain free group automorphisms and the asymptotic growth of the index of the intersection of all index n subgroups of a group. Finally, the PI hopes to promote and expand the exciting new topic of `invariant random subgroups' of Lie groups, which he and Abert et al used to control the growth of Betti numbers of higher rank locally symmetric spaces.The study of geometric shapes of 2 and 3 dimensions has rapidly evolved in the past 40 years, partly due to revolutionary work of William Thurston in the 1970's. Thurston was one of the first to realize the particular utility of 'geometric degenerations' in these low dimensions. A geometric degeneration can be thought of as a deformation of a shape that possibly changes its type at the end: for instance, a sphere with a shrinking radius degenerates to a point and a vertical cylinder with fixed height and shrinking circumference degenerates to a line segment. The goal of the proposed project is to use geometric degenerations to analyze a variety of low-dimensional shapes, especially 3-dimensional manifolds, of which our universe is an example.