Hierarchy Theory for Automorphism and Outer Automorphism Groups of Free Groups

对对称性的研究可以追溯到古代--关于柏拉图固体的经典定理可以在欧几里得的《元素》中找到。中世纪的工匠们用美丽的平面对称性装饰阿尔罕布拉宫的墙壁，在现代语言中，也被称为二维结晶学群。抽象有很大的力量，对称性的研究在19世纪中叶数学家们提出了一个抽象的群体概念时出现了飞跃。这些进展使科学的应用得以蓬勃发展，例如使用三维晶体学基团来探索晶体阵列的原子结构。现代几何群论是对晶体群研究的广泛抽象。在该奖项下完成的研究将通过关注当前知识前沿的一类群，即自由群的自同构群和外自同构群，来促进我们对几何群论的理解。这些群编码网络之间的对称性和其他关系，数学家称之为图。该奖项的工作将涉及构建图形之间关系和对称性的新模型，导致几何群论中新定理的发现和证明。其中一些工作将由研究生在首席研究员的建议下进行，针对初学研究生的数学专著将被制作出来，以帮助他们的培训和更广泛的数学界使用。该奖项下的研究活动(预计是与纽约大学的Michael Handel联合开展的)将集中研究群AUT(F)和OUT(F)的大规模几何，有限秩自由群F的自同构群和外自同构群。在大比例尺几何学中，人们通过其对各种几何形状的作用来研究一组几何图形，重点不是这些几何图形的小尺度特征，而是大尺度特征的差异和共性。在这个奖项下，AUT(F)和OUT(F)将通过发展这些群的层次理论来研究，类似于如此成功地用于研究映射类曲面群的大规模几何的层次理论。构建一个层次理论至少需要群及其各种子群在双曲测地度量空间上的作用。关于OUT(F)的层次结构，有些事情是已知的，包括表示层次结构顶部的操作。但我们对这一主题的了解仍然存在许多巨大的差距。为了填补Out(F)层次结构中间的空白，本奖项下要追求的一个策略是为F的自由因子A构建代表Aut(A)层次结构顶部的动作。另一个策略将是构建编码Dehn扭曲行为的动作，这是填补Out(F)层次结构底部空白所必需的。中期应用有望揭示字的几何性质和OUT(F)的共轭问题。更长期的应用可能包括揭示关于Out(F)的有限渐近维度和准等距刚性的深层猜想。在该奖项下完成的研究生工作有望将层次方法扩展到自由群的外自同构群之外，以包括更广泛类别的外自同构群。

The study of symmetry dates to ancient times - classic theorems on Platonic solids are found in Euclid's "Elements." Medieval artisans decorated the walls of the Alhambra palace with beautiful depictions of planar symmetries also known, in modern language, as 2-dimensional crystallographic groups. There is great power in abstraction, and the study of symmetry leaped forward in the mid 19th century when mathematicians formulated the abstract concept of a group. These advances enabled a flowering of applications to science, such as the use of 3-dimensional crystallographic groups to explore the atomic structure of crystalline arrays. Modern geometric group theory is a vast abstraction of the study of crystallographic groups. The research accomplished under this award will advance our understanding of geometric group theory by focusing on a class of groups at the current frontier of knowledge, namely the automorphism and outer automorphism groups of free groups. These groups encode symmetries and other relations amongst networks which mathematicians call graphs. Work under this award will involve construction of new models of relations and symmetries amongst graphs, leading to the discovery and proof of new theorems in geometric group theory. Some of this work will be carried out by graduate students under the advisement of the principal investigator, and mathematical monographs aimed at beginning graduate students will be produced to aid in their training and for use by the broader mathematical community.Research activity under this award (expected to be joint work with Michael Handel of CUNY) will focus on studying the large scale geometry of the groups Aut(F) and Out(F), the automorphism and outer automorphism groups of a free group F of finite rank. In large scale geometry one studies a group by its actions on various geometries, focusing not on small scale features of those geometries, but instead on differences and commonalities of large scale features. Under this award Aut(F) and Out(F) will be studied by developing hierarchy theories for these groups, in analogy to hierarchy theories used so successfully to study the large scale geometry of mapping class groups of surfaces. Constructing a hierarchy theory requires, at a minimum, actions of the group and various of its subgroups on hyperbolic geodesic metric spaces. Some things are known already about a hierarchy for Out(F), including actions representing the top of a hierarchy. But there remain many large gaps in our knowledge of this topic. To plug gaps in the middle of an Out(F) hierarchy, one strategy to be pursued under this award will be constructing actions representing the top of Aut(A) hierarchies for free factors A of F. Another strategy will be constructing actions that encode Dehn twist behavior, needed to plug gaps at the bottom of an Out(F) hierarchy. Medium term applications are expected to reveal geometric properties of word and conjugacy problems for Out(F). Longer term applications may include shedding light on deep conjectures about finite asymptotic dimension and quasi-isometric rigidity of Out(F). Graduate work done under this award is expected to extend hierarchy methods beyond outer automorphism groups of free groups to include broader categories of outer automorphism groups.