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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759094
• 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（编码对象上的所有指标）及其对称组。该建议描述了对两个这样的变形空间的研究：Culler-Vogtmann外层空间和Teichmuller空间。外层空间是一个简单的复合物（减去某些面），编码了固定基本组的所有加权图。我研究外太空，重点是两个相互联系的主题：（1）一个通过研究外层空间的地球化学如何有效地变形了此类加权图，以及（2）当人们反复将自动形态应用于自由组元素时，会发生什么，即渐近不变。 特征向量和特征值是基质组设置中渐近不变的经典示例。而且，就像在这种情况下一样，这些不变性在理解发生的事情中反复应用自动形态和指标的有效变形时起着至关重要的作用，再次编码在外层空间的地球上。我描述了一个克服实质性障碍的计划，以建立和理解外太空某些大地测量学有用的动力学系统。同时，我回答了一个问题，询问哪些渐近不变剂在随机步行和熵感官中最常见。 我们在该计划中大大扩展的技术的新颖性在其离散地编码地理学，以一种利用和照亮其与不变的关系的方式。尽管现在开始探索更广泛的了解外层空间的应用，但它们显然是广泛的。加权图在生物学，人工智能和更通用的计算机科学等各种环境中出现。外太空甚至与代数几何形状，特别是热带几何形状有关。 Teichmuller空间没有编码加权图，而是在固定有限的表面上编码双曲线指标。我们在Teichmuller空间上的工作是提供瑟斯顿紧凑型的替代证明。我们方法的好处是两个方面。首先，我们构建叶子密切接近双曲线指标。其次，我们的方法可用于其他设置，例如在表面上凸出的凸出结构的空间中。我计划培训11个HQP，专注于在广泛活跃的研究领域的教学（高度转移）技能，并通过帮助他们在与新的合作者结识新的合作者的同时，通过帮助他们进行交流和背景化的数学来沟通和背景化，从而为成功的数学职业做好了整体准备。为研究生提供了外层空间的项目，因为他们所学的技能不仅可以进一步研究外太空，而且还可以使他们准备参加模仿用于研究外层空间的方法的最新趋势，以及在研究外层空间的趋势和自由组的自动形态群体。研究其他群体，并研究建模天然系统的图形，例如系统发育树或技术系统，例如神经网络。