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

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。本文描述了两个这样的变形空间：Culler-Vogtmann外空间和Teichmuller空间。外空间是一个简单复（减去一些面）编码一个固定基本群的所有加权图。我研究外层空间的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，通过研究外层空间的测地线，以及(2)当一个人反复应用自同构到一个自由群元素，即渐近不变量时，会发生什么。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，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起着至关重要的作用，度量的有效变形再次编码在外太空的测地线中。我描述了一个计划，以克服在外层空间建立和理解某些测地线的有用动力系统的实质性障碍。同时，我还回答了在随机漫步和熵的意义下，自由群自同构中最常见的渐近不变量是什么。我们的技术的新颖性，我们在程序中大大扩展，在他们的离散编码测地线，以一种方式利用和阐明它们与不变量的关系。虽然了解外层空间的更广泛的应用才刚刚开始探索，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。外太空甚至与代数几何有关，特别是热带几何。代替编码加权图，Teichmuller空间编码固定有限曲面上的双曲度量。我们在Teichmuller空间上的工作是给出其瑟斯顿紧化的另一种证明。我们的方法的好处是双重的。首先，我们构造了近似于双曲度量的叶形。其次，我们的方法可以用于其他设置，例如在曲面上紧化凸投影结构的空间。我计划培养11名hqp，重点是在一个广泛活跃的研究领域教授（高度可转移的）技能，并通过帮助hqp学习沟通和将数学背景化，同时结识新的合作者，为成功的数学事业做好全面准备。研究生被赋予外太空项目，因为他们学到的技能不仅可以进一步研究外太空，而且可以帮助他们参与最近的模仿方法的趋势，这些方法用于研究外太空和自由群的自同构群，研究其他组，以及研究自然系统建模的图，如系统发育树，或技术系统，如神经网络。

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.