Outer automorphisms of free groups, outer space, and related deformation spaces

自由群、外层空间和相关变形空间的外自同构

• 批准号: RGPIN-2019-04318
• 负责人: Pfaff, Catherine
• 金额: $ 1.17万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713915
• 关键词: Outer; automorphisms; free; groups; outer

One of the most beautiful interplays in mathematics is between a deformation space (encoding all metrics on an object) and its symmetry group. This proposal describes a study of two such deformation spaces: Culler-Vogtmann outer space and Teichmuller space. Outer space is a simplicial complex (minus some faces) encoding all weighted graphs of a fixed fundamental group. I study outer space with a focus on two interconnected themes: (1) how one efficiently deforms such weighted graphs, via studying geodesics in outer space, and (2) what happens when one repeatedly applies an automorphism to a free group element, i.e. the asymptotic invariants. Eigenvectors and eigenvalues are classical examples of asymptotic invariants in a matrix group setting. And, just as in that setting, these invariants play a crucial role in understanding both what happens as one repeatedly applies the automorphism and the efficient deformation of metrics, again encoded in the geodesics of outer space. I describe a plan for overcoming substantial obstacles to build and understand a useful dynamical system of certain geodesics in outer space. In conjunction, I give answers to the question asking which asymptotic invariants are most common for free group automorphisms, both in random walk and entropy senses. The novelty of our techniques, which we greatly expand in the program, are in their discretely codifying geodesics, in a manner utilizing and illuminating their relationships to the invariants. While more broad applications of understanding outer space are just now beginning to be explored, they are clearly extensive. Weighted graphs arise in such diverse settings as biology, artificial intelligence, and more general computer science. Outer space even relates to algebraic geometry, specifically tropical geometry. Instead of encoding weighted graphs, Teichmuller space encodes hyperbolic metrics on a fixed finite surface. Our work on Teichmuller space is in giving an alternate proof of its Thurston compactification. The benefit of our approach is two-fold. First, we construct foliations closely approximating hyperbolic metrics. Second, our methods may be used in other settings, such as in compactifiying the space of convex projective structures on a surface. I plan to train 11 HQPs, with focus on teaching (highly transferrable) skills in an extensively active research area and overall preparing HQPs for successful mathematics careers by helping them learn to communicate and contextualize their mathematics, while meeting new collaborators. The graduate students are given outer space projects, as the skills they learn will not only allow further study of outer space, but will prepare them to participate in recent trends of mimicking methods used to study outer space and the automorphism group of the free group in studying other groups, and in studying graphs modeling natural systems, such as phylogenetic trees, or technological systems, such as neural networks.

数学中最美丽的相互作用之一是变形空间(对一个对象的所有度量进行编码)和它的对称群之间的相互作用。这一建议描述了两个这样的变形空间的研究：卡勒-沃格特曼外层空间和泰希米勒空间。 外层空间是编码固定基本群的所有赋权图的单纯复数(减去某些面)。我对外层空间的研究集中在两个相互关联的主题上：(1)如何通过研究外层空间的测地线来有效地变形这种加权图，以及(2)当一个自由群元素重复应用自同构时会发生什么，即渐近不变量。 特征向量和特征值是矩阵群设置中渐近不变量的经典例子。而且，就像在那个环境中一样，这些不变量在理解一个人重复应用自同构和度量值的有效形变时都发挥了关键作用，度量值同样编码在外层空间的测地线中。 我描述了一项克服重大障碍以建立和理解外层空间某些测地线的有用动力系统的计划。同时，我给出了关于自由群自同构最常见的渐近不变量的问题的答案，无论是在随机游动意义下还是在熵意义下。 我们在程序中大大扩展的技术的新奇之处在于，它们以一种利用和阐明它们与不变量的关系的方式，离散地对测地线进行编码。 虽然了解外层空间的更广泛的应用现在才刚刚开始探索，但它们显然是广泛的。加权图出现在生物学、人工智能和更一般的计算机科学等不同的环境中。外层空间甚至涉及到代数几何，特别是热带几何。 Teichmuller空间不对加权图进行编码，而是对固定有限曲面上的双曲度量进行编码。我们在TeichMuller空间上的工作是给出它的瑟斯顿紧化的另一种证明。我们的方法有两方面的好处。首先，我们构造了接近于双曲度量的叶层。其次，我们的方法也可用于其他情况，如紧致曲面上的凸射影结构的空间。 我计划培训11名HQP，重点是在一个广泛活跃的研究领域传授(高度可移植的)技能，并通过帮助他们学习交流和将他们的数学联系起来，同时结识新的合作者，为成功的数学职业生涯做好总体准备。研究生被赋予外层空间项目，因为他们学到的技能不仅将使他们能够进一步研究外层空间，而且将使他们准备参与最近的趋势，模仿用于研究外层空间的方法和自由群体的自同构群研究其他群体，并研究对自然系统(如系统发育树)或技术系统(如神经网络)进行建模的图形。