Graphs, Trees and Geometric Group Theory

群论中的一个基本技巧是通过考察群在拓扑空间上的作用来研究群。 如果把注意力限制在具有强几何性质的空间（如度量树或CAT（0）空间）上，人们通常可以使G的作用集 变成一个拓扑空间本身，称为变形空间，具有良好的拓扑性质。 G的自同构群通过扭曲每个作用量而作用在变形空间上，得到一个新的作用量。 在这个项目中，我们通过研究它们在变形空间上的作用来研究各类群的自同构群。 对于自由群，最初由PI和M定义的相关变形空间。卡勒，被称为外太空。 在这个问题上的一个相当新的发展是，森田利用Kontsevich发现的外层空间和某个无限维辛李代数之间的联系来检测Out（F_n）的上循环，Out（F_n）是秩为n的自由群的外自同构群。 PI和J.Conant将这些上圈重新解释为Out（F_n）作用于外层空间商上的圈，并计划利用它们来研究Out（F_n）的不稳定上同调。 该项目的第二个目标是与R. Charney，是开发一个类似的外太空直角阿廷群，一类，其中包括自由群和自由阿贝尔群。 第三部分是由PI以前关于Out（F_n）和Aut（F_n）同调稳定性的工作所激发的。 证明这些稳定性结果的技术适用于其他序列的低维拓扑相关的群体，PI将进一步研究这些应用。其他正在进行的项目包括系统发育树空间的研究和Out（F_n）的刚性性质的调查。 在数学和科学中，有一种现象是，通过将复杂的结构所包含的信息以图形和树的形式进行编码，往往可以更容易地理解复杂的结构。 一个例子是使用图和树来描述一个被称为群的代数对象可能分裂成更简单的片段。当存在许多可能的分裂时，可以构造测量模糊性的变形空间。 在这个项目中，我们通过考虑群的对称性在这种变形空间上的作用来研究群。

A fundamental technique in group theory is to study groups by examining their actions on topological spaces. If one restricts attention to spaces with strong geometric properties (such as metric trees or CAT(0) spaces), one can often make the set of actions of G into a topological space itself, called a deformation space, with good topological properties. The automorphism group of G acts on the deformation space by twisting each action to get a new action. In this project we study automorphism groups of various classes of groups by studying their actions on deformation spaces. For free groups, the relevant deformation space, originally defined by the PI and M. Culler, is known as Outer space. A fairly recent development in this subject is that Morita used a connection found by Kontsevich between Outer space and a certain infinite-dimensional symplectic Lie algebra to detect cocycles for Out(F_n), the group of outer automorphisms of the free group of rank n. The PI and J. Conant have reinterpreted these cocycles as cycles on the quotient of Outer space by the action of Out(F_n), and plan to use these to study the unstable cohomology of Out(F_n). A second goal of this project, joint with R. Charney, is to develop an analog of Outer space for right-angled Artin groups, a class which includes both free groups and free abelian groups. A third component is motivated by previous work of the PI on stability of the homology of Out(F_n) and Aut(F_n). Techniques developed for proving these stability results apply to other sequences of groups related to low-dimensional topology, and the PI will further investigate these applications. Other ongoing projects include a study of the space of phylogenetic trees and an investigation of rigidity properties of Out(F_n). A phenomenon which occurs throughout mathematics and the sciences is that complicated structures can often be understood more easily by codifying the information they contain in terms of graphs and trees. On example of this is the use of graphs and trees to describe the possible splittings of an algebraic object called a group into simpler pieces. When there are many possible splittings, one can construct deformation spaces which measure the ambiguity. In this project we study groups by considering actions of the symmetries of the group on such deformation spaces.