Topological aspects of infinite group theory

在过去的一百年里，一个基本的几何洞见是对称性(广义解释)支配着大量的现象。即使是许多看似非几何的物体也可以通过对称性来研究，这给了它们一种隐藏的几何图形，可以用来研究以其他方式无法回答的问题。本项目将关注这种对称性的拓扑方面。在这里，拓扑学是指研究在弯曲和拉伸下不变的空间的大规模性质；例如，高维洞的存在。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、会议组织和说明性写作，目的是扩大在不同水平上对数学的参与。研究人员将在映射类群、自由群的自同构群和算术群的拓扑研究方面取得进展，着眼于在几何拓扑学和几何群论的研究对象与代数几何和表示论之间架起桥梁。拟议的工作有四个主要方向。首先，PI将稳定同调的结果推广到有限指标子群。在第二个方面，PI将在自由群的类群和自同构群何时与紧致Kahler流形的基本群可公化的问题上取得进展，从而在几何拓扑和代数几何中的群之间建立了一种新的联系。在第三章中，PI将证明Torelli群的同调的一种形式的表示稳定性。最后，PI将澄清算术群的不稳定同调和斯坦伯格表示之间的关系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.