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空间序列通常是有用的。PI提出了五个项目，将这种渐近方法应用于拓扑，几何和群论。 首先，PI将研究闭双曲三维流形的大尺度几何，其基本群可以由固定数量的元素生成。其次，PI将研究凸余紧体的代数和几何极限，以及它们与3-流形边界的同胚向其内部的扩张的关系。 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数n子群的交集的指数的渐近增长。 最后，PI希望促进和扩大令人兴奋的新课题'不变随机子群'的李群，他和阿伯特等人用来控制增长的贝蒂数的高阶局部对称spaces.The研究的几何形状的2和3维已经迅速发展，在过去的40年，部分原因是革命性的工作威廉瑟斯顿在20世纪70年代。 瑟斯顿是第一个意识到“几何退化”在这些低维度的特殊效用。 几何退化可以被认为是一个形状的变形，它可能在最后改变它的类型：例如，一个半径缩小的球体退化为一个点，一个高度固定且周长缩小的垂直圆柱体退化为一条线段。 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.