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（对对象上的所有度量进行编码）与其对称群之间的相互作用。该提案描述了对两个此类变形空间的研究：Culler-Vogtmann 外层空间和 Teichmuller 空间。 外层空间是一个单纯复形（减去一些面），编码固定基本群的所有加权图。我研究外层空间的重点是两个相互关联的主题：（1）如何通过研究外层空间的测地线来有效地变形此类加权图，以及（2）当一个人重复将自同构应用于自由群元素时会发生什么，即渐近不变量。 特征向量和特征值是矩阵组设置中渐近不变量的经典示例。而且，正如在这种情况下一样，这些不变量在理解重复应用自同构和度量的有效变形时发生的情况方面发挥着至关重要的作用，再次编码在外层空间的测地线中。 我描述了一个克服重大障碍的计划，以建立和理解外层空间某些测地线的有用动力系统。结合起来，我给出了以下问题的答案：在随机游走和熵意义上，自由群自同构中哪些渐近不变量是最常见的。 我们在程序中大大扩展的技术的新颖性在于它们离散地编码测地线，以利用和阐明它们与不变量的关系的方式。 虽然了解外层空间的更广泛应用刚刚开始探索，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。外层空间甚至与代数几何，特别是热带几何有关。 Teichmuller 空间不是对加权图进行编码，而是在固定的有限表面上对双曲度量进行编码。我们对泰希米勒空间的工作是给出瑟斯顿紧致化的另一种证明。我们的方法有两个好处。首先，我们构建非常接近双曲度量的叶状结构。其次，我们的方法可以用于其他设置，例如压缩表面上凸射影结构的空间。 我计划培训 11 名 HQP，重点是在广泛活跃的研究领域教授（高度可转移的）技能，并通过帮助 HQP 学习交流和将其数学情境化，同时结识新的合作者，为 HQP 成功的数学职业生涯做好全面准备。研究生将接受外太空项目，因为他们学到的技能不仅可以让他们进一步研究外太空，而且可以让他们为参与研究外太空和自由群的自同构群的模仿方法的最新趋势做好准备。研究其他群体，以及研究对自然系统进行建模的图形，例如系统发育树，或技术系统，例如神经网络。