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（编码对象上的所有度量）与其对称群之间的相互作用。该提案描述了对两个这样的变形空间的研究：Culler-Vogtmann外部空间和Teichmuller空间。 外层空间是一个单纯复形（减去一些面），编码一个固定基本群的所有加权图。我研究外层空间，重点是两个相互关联的主题：（1）如何有效地变形这样的加权图，通过研究测地线在外层空间，和（2）会发生什么，当一个反复应用自同构到一个自由群元素，即渐近不变量。 特征向量和特征值是矩阵群中渐近不变量的经典例子。而且，就像在这种情况下一样，这些不变量在理解重复应用自同构和度量的有效变形时会发生什么方面发挥着至关重要的作用，这些变形也被编码在外层空间的测地线中。 我描述了一个计划，克服实质性的障碍，建立和理解一个有用的动力系统的某些测地线在外层空间。在结合，我给的问题，问渐近不变量是最常见的自由群自同构，无论是在随机游走和熵的意义上的答案。 我们的技术的新奇，我们大大扩大了计划，是在他们离散编纂测地线，在某种程度上利用和照亮他们的关系的不变量。 虽然现在才开始探索了解外层空间的更广泛的应用，但这些应用显然是广泛的。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有关，特别是热带几何。 而不是编码加权图，Teichmuller空间编码双曲度量在一个固定的有限表面。我们在Teichmuller空间上的工作是给出其Thurston紧化的另一种证明。我们的方法有两方面的好处。首先，我们构造接近双曲度量的叶理。其次，我们的方法可以用于其他设置，如在紧化的凸投影结构的空间上的表面。 我计划培训11名HQP，重点是在一个广泛活跃的研究领域教授（高度可转移）技能，并通过帮助他们学习沟通和情境化他们的数学，同时满足新的合作者，为成功的数学职业做好全面的准备。研究生获得外层空间项目，因为他们学到的技能不仅可以进一步研究外层空间，而且将使他们准备参与最近的趋势，即模仿用于研究外层空间的方法和自由组的自同构组研究其他组，并研究自然系统的图形建模，如系统发育树，或技术系统，如神经网络。