Automorphism Groups and Morse Boundaries

摘要奖：DMS 1607616，首席研究员：Ruth Charney数学用于对物理系统建模和分析数据集。这些模型越来越多地采用几何对象的形式。这个项目专注于一类几何对象，这些对象在包括机器人学和遗传学在内的许多背景下作为模型出现。这些模型及其对称群的几何性质反映在物理系统的性质中。问题可能涉及局部几何体(在特定点附近发生的事情)或大尺度几何体(从远处观察对象的结构)。这个项目涉及大规模的几何学。通过为所讨论的几何引入“边界”的新概念，我们能够识别和量化某些类型的行为，即所谓的双曲线行为。测地度量空间的边界在双曲群的研究中起着重要的作用，例如在证明刚性定理和动力学性质方面。也可以为CAT(0)空间定义边界，但它们不是拟等距不变的，因此没有给出CAT(0)群的定义良好的边界。在最近的工作中，首席研究员和Sultan为CAT(0)空间引入了一个新的边界，称为Morse边界，它是准等距不变的，其行为更像双曲空间的边界。随后，Cordes推广了这一结构，得到了任意真测地度量空间的Morse边界。这为研究大类群，如非单线性双曲群提供了一个潜在的强大的新工具。本项目的第一部分将调查这些边界的性质及其对有关群体的影响。该项目的第二部分涉及直角Artin群(RAG)的自同构群。自由群、映射类群和线性群的自同构群有许多共同的性质。RAG的自同构群为研究这些共性提供了一个背景。由于Teichmuller空间一直是研究映射类群的中心，而Culler和Vogtmann的外空间一直是研究自由群自同构群的基本工具。同样，曲面的曲线复形的类比也被引入到自由群中，并被证明是双曲线的。这个项目的第二部分试图为RAG找到一个类似的外层空间，并研究所有RAG的自由因子和自由分裂复合体的推广。

AbstractAward: DMS 1607616, Principal Investigator: Ruth CharneyMathematics is used to model physical systems and to analyze data sets. Increasingly, these models take the form of geometric objects. This project focuses on a class of geometric objects that arise as models in a number of contexts, including robotics and genetics. Geometric properties of these models and their symmetry groups are reflected in properties of the physical systems. Questions may involve either local geometry (what happens near a particular point) or large-scale geometry (the structure of the object viewed from a distance). This project concerns large-scale geometry. By introducing a new notion of a "boundary" for the geometries in question, we are able to identify and quantify certain types of behavior, known as hyperbolic behavior. The study of these boundaries, and their implications for the geometry and symmetry of the objects in question, is the main theme of this project.Boundaries of geodesic metric spaces have played an important role in the study of hyperbolic groups, for example in proving rigidity theorems and dynamical properties. Boundaries can also be defined for CAT(0) spaces, however they are not quasi-isometry invariant, hence do not give a well-defined boundary for a CAT(0) group. In recent work, the principal investigator and Sultan introduced a new boundary, called the Morse boundary, for CAT(0) spaces which is quasi-isometry invariant and behaves more like the boundary of a hyperbolic space. Subsequently, Cordes generalized this construction to obtain a Morse boundary for any proper geodesic metric space. This offers a potentially powerful new tool for studying large classes of groups, such as acylindrically hyperbolic groups. The first part of this project will investigate properties of these boundaries and their implications for the groups in question. The second part of the project concerns automorphism groups of right-angled Artin groups (RAAGs). Automorphism groups of free groups, mapping class groups, and linear groups have many properties in common. Automorphism groups of RAAGs give a context in which to study these commonalities. As Teichmuller space has been central to the study of mapping class groups, Culler and Vogtmann's Outer space has been a fundamental tool in the study of automorphism groups of free groups. Similarly, analogues of the curve complex of a surface have been introduced for free groups and shown to be hyperbolic. The second part of this project seeks to find an analogous Outer Space for RAAGs and to study generalizations of the free factor and free splitting complexes for all RAAGs.