Asymptotic methods in groups and locally symmetric spaces

通过几何限制来理解一类或一系列空间通常很有用。PI提出了将这种渐近方法应用于拓扑学、几何学和群论的五个项目。首先，PI将研究闭双曲3-流形的大规模几何，其基本群可以由固定数量的元素生成。其次，PI将研究凸余紧手柄的代数极限和几何极限，以及它们与三维流形边界的同胚延展到其内部的关系。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以及群的所有指标n个子群的交的指数的渐近增长性。最后，PI希望推广和扩展他和Abert等人用来控制高阶局部对称空间的Betti数的增长的令人兴奋的新主题--李群的不变随机子群。在过去的40年里，对2维和3维几何形状的研究迅速发展，部分原因是威廉·瑟斯顿在1970年的《S》一书中所做的革命性工作。瑟斯顿是最先认识到在这些低维空间中的特殊用途的‘几何退化’的人之一。几何退化可以被认为是形状的变形，该变形可能在末端改变其类型：例如，具有收缩半径的球体退化为点，具有固定高度和收缩圆周的垂直圆柱体退化为线段。拟议项目的目标是使用几何简并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.