Asymptotic invariants of groups and subgroups

群和子群的渐近不变量

• 批准号: 1006345
• 负责人: Denis Osin
• 金额: $ 12.48万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006345&HistoricalAwards=false
• 关键词: Asymptotic; invariants; groups; subgroups

This proposal consists of three parts devoted to relatively hyperbolic groups, their generalizations, and asymptotic invariants of residually finite groups. The first part of this proposal is inspired by some important purely algebraic problems, which can be solved using geometric small cancellation theory over relatively hyperbolic groups developed by the PI. The second part proposes a generalization of relative hyperbolicity, which encompasses many interesting groups acting "nicely" on hyperbolic spaces (e.g., mapping class groups, outer automorphism groups of free groups, fundamental groups of graphs of groups, etc.) The third part of the proposal is inspired by important open problems about three asymptotic invariants of residually finite groups: rank gradient, L2-betti numbers, and cost of group actions.The general idea behind this project is to develop new methods to study geometry and asymptotic invariants of groups and subgroups. The proposed approach is mostly based on recent advances in the theory of relatively hyperbolic groups. Lots of motivation for this project come from other branches of mathematics such as non-commutative geometry, low-dimensional topology, and topological dynamics. If successful, the project will help to create new powerful tools in geometric group theory. On the other hand, solution the proposed problems will likely have strong impact on other areas of mathematics.

这个建议包括三个部分致力于相对双曲群，他们的推广，和剩余有限群的渐近不变量。 这个建议的第一部分是受到一些重要的纯代数问题的启发，这些问题可以使用PI开发的相对双曲群上的几何小消去理论来解决。 第二部分提出了相对双曲性的一个推广，它包含了许多有趣的群在双曲空间上“很好地”起作用（例如，映射类群、自由群的外自同构群、群的图的基本群等）该计划的第三部分是受到关于剩余有限群的三个渐近不变量的重要公开问题的启发：秩梯度，L2-betti数和群作用的代价。该项目背后的总体思想是发展新的方法来研究群和子群的几何和渐近不变量。所提出的方法主要是基于相对双曲群理论的最新进展。这个项目的许多动机来自其他数学分支，如非交换几何，低维拓扑和拓扑动力学。如果成功，该项目将有助于在几何群论中创建新的强大工具。另一方面，这些问题的解决可能会对数学的其他领域产生重大影响。