CAREER: Asymptotic invariants of residually finite groups

Abert将研究剩余有限群的子群格上自然不变量的渐近行为。这些不变量的例子有等级、成本、Betti数、Heegaard亏格、顺应性、谱间隙、有界世代和围长。这个项目的历史背景是子群的增长和超定群(Lubotzky，Segal，Shalev和Wilson)，有限表示群和拓扑(Luck和Lackenby)，轨道等价(Weiss，Popa和Furman)和自相似群(Grigorchuk和Nekrasevich)。如果一个群的有限指数的子群的交是平凡的，则称它为剩余有限群。这意味着有限图像近似于群结构。重要的例子是有限生成的线性群，特别是算术群。感兴趣的核心对象是有限指标子群的降序链。链中有限陪集作用的组合学、相应陪集树边界上的保测作用的动力学和离散群的结构之间存在着有趣的相互作用。这种相互作用使人们可以应用可测群理论中的刚性定理，并得到新的图论结果。拟议的活动还与传递图上的渗流和3-流形理论有关。群论是一种古老的核心数学原理，诞生于19世纪初。任意物体的对称性集合形成一个群，因此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。Abert将研究剩余有限群；这些是有限群和无限群的自然交汇点。拟议的活动位于群论、图论和动力学的十字路口，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，通过创造性的问题解决让他们接触到他的部分研究。该项目还与芝加哥大学维格雷项目进行了协调。最终目标是在研究性学习层面上形成研究性学习的核心模式。

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.