Topological aspects of infinite group theory

在过去的一百年里，几何学的基本观点之一是对称性（广义上的解释）支配着大量的现象。 甚至许多看似非几何的物体也可以通过对称性进行研究，这给了它们一种隐藏的几何结构，可以用来研究无法用其他方法回答的问题。 这个项目将集中在这种对称性的拓扑方面。 这里拓扑学是指研究在弯曲和拉伸下不变的空间的大尺度性质;例如，高维孔的存在。 PI和他的博士生将调查这一领域的一系列问题。 PI还将继续参与高中外展，会议组织和论文写作，旨在扩大各级数学的参与。调查人员将取得进展的拓扑结构的研究映射类群体，自同构群的自由群体，和算术群体的眼睛对建立桥梁之间的研究对象的几何拓扑结构和几何群论一方面，代数几何和代表性理论的其他。 拟议的工作有四个主要方向。 在第一，PI将扩大结果稳定的同调有限指数子群。 在第二，PI将取得进展的开放性问题时，是映射类群和自同构群的自由团体可扩展到基本组的紧凑的卡勒流形，从而开发了一种新的连接之间产生的群体几何拓扑和代数几何。 在第三部分中，PI将证明Torelli群的同调的表示稳定性。最后，PI将阐明算术群的不稳定同调和Steinberg表示之间的关系。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the fundamental geometric insights of the last hundred years is that symmetry (broadly interpreted) governs a vast array of phenomena. Even many seemingly non-geometric objects are amenable to study via symmetry, giving them a sort of hidden geometry that can be used to study questions that cannot be answered otherwise. This project will focus on topological aspects of this symmetry. Here topology refers to the study of large-scale properties of spaces that are invariant under bending and stretching; for instance, the presence of high-dimensional holes. The PI and his PhD students will investigate an array of questions in this area. The PI will also continue his engagement in high school outreach, conference organization, and expository writing with the aim of broadening participation in mathematics at a variety of levels. The investigator will make advances in the study of the topology of mapping class groups, automorphism groups of free groups, and arithmetic groups with an eye toward building bridges between objects of study in geometric topology and geometric group theory on the one hand, and algebraic geometry and representation theory on the other. The proposed work has four main directions. In the first, the PI will extend results on stable homology to finite-index subgroups. In the second, the PI will make advances in the open question when are mapping class groups and automorphism groups of free groups commensurable to the fundamental group of a compact Kahler manifold, thus developing a novel connection between groups arising in geometric topology and in algebraic geometry. In the third, the PI will prove a version of representation stability for the homology of the Torelli group. Finally, the PI will clarify the relations between the unstable homology of arithmetic groups, and the Steinberg representation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.