CAREER: Asymptotic invariants of residually finite groups

职业：残差有限群的渐近不变量

• 批准号: 0847387
• 负责人: Miklos Abert
• 金额: $ 41.29万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-02-01 至 2010-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0847387&HistoricalAwards=false
• 关键词: CAREER; Asymptotic; invariants; residually; finite

Abert will investigate the asymptotic behavior of natural invariants on the subgroup lattice of residually finite groups. Examples for such invariants are rank, cost, Betti numbers, Heegaard genus, amenability, spectral gap, bounded generation and girth. The historical background of the project is subgroup growth and profinite groups (Lubotzky, Segal, Shalev and Wilson), finitely presented groups and topology (Luck and Lackenby), orbit equivalence (Weiss, Popa and Furman) and self-similar groups (Grigorchuk and Nekrasevich).A group is called residually finite, if the intersection of its subgroups of finite index is trivial. This means that finite images approximate the group structure. Important examples are finitely generated linear groups, and specifically, arithmetic groups. The core object of interest is a descending chain of finite index subgroups. There is an interesting interplay between the combinatorics of the finite coset actions in the chain, the dynamics of the measure preserving action on the boundary of the corresponding coset tree and the structure of the discrete group. This interplay allows one e.g. to apply rigidity theorems in measurable group theory and get new graph theoretical results. The proposed activity also has connections to percolation on transitive graphs and the theory of 3-manifolds.Group theory is an old and central mathematical principle, born in the early 19th century. The set of symmetries of an arbitrary object forms a group, so groups arise virtually in all areas in mathematics and also in certain parts of physics and chemistry. Abert will study residually finite groups; these are natural meeting points of finite and infinite groups. The proposed activity lies at the crossroads of group theory, graph theory and dynamics and has strong connections to certain areas in probability theory and topology; as such, it is highly interdisciplinary. As part of the project, Abert will work with gifted undergraduate and graduate students and expose them to parts of his research through creative problem solving. The project is also coordinatedwith the University of Chicago VIGRE Program. The ultimate goal is to work out a core format for inquiry based learning on research level.

Abert将研究剩余有限群的子群格上的自然不变量的渐近行为。这样的不变量的例子是秩，成本，贝蒂数，Heegaard属，顺从性，频谱间隙，有界生成和周长。该项目的历史背景是子群增长和profinite群（Lubotzky，Segal，Shalev和Wilson），群的可表示性和拓扑（Luck和Lackenby），轨道等价（韦斯，Popa和Furman）和自相似群（Grigorchuk和Nekrasevich）。这意味着有限像近似于群结构。重要的例子是代数生成的线性群，特别是算术群。感兴趣的核心对象是有限指数子群的下降链。有一个有趣的组合之间的相互作用的有限陪集行动的链，动力学的措施保持行动的边界上相应的陪集树和结构的离散组。这种相互作用允许一个例如应用刚性定理在可测群理论，并得到新的图论结果。建议的活动也有连接到渗流的传递图和理论的3-manifolds.Group理论是一个古老的和中心的数学原则，诞生于19世纪初世纪。一个任意对象的对称集合形成一个群，所以群实际上出现在数学的所有领域，也出现在物理和化学的某些部分。阿伯特将研究剩余有限群;这些是自然的交汇点有限和无限的群体。拟议的活动位于群论，图论和动力学的十字路口，并与概率论和拓扑学的某些领域有很强的联系;因此，它是高度跨学科的。作为该项目的一部分，阿伯特将与有天赋的本科生和研究生合作，并通过创造性的解决问题使他们接触到他的部分研究。该项目还与芝加哥大学的VIGRE项目进行了协调。研究性学习的最终目标是制定出研究性学习的核心模式。