Graphs, Trees and Geometric Group Theory

群论的一项基本技术是通过考察群在拓扑空间上的行为来研究群。如果将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，则通常可以将G的动作集本身变成具有良好拓扑性质的拓扑空间，称为变形空间。G的自同构群通过扭转每个动作来作用于变形空间，从而得到一个新的动作。本课题通过研究群在变形空间上的作用来研究各种群的自同构群。对于自由群，最初由PI和M. Culler定义的相关变形空间称为外空间。在这一课题中，Morita利用Kontsevich发现的外空间与某个无限维辛李代数之间的联系来检测n秩自由群的外自同构群Out（F_n）的环。PI和J. Conant通过Out（F_n）的作用将这些环重新解释为外空间商上的环，并计划利用这些环来研究Out（F_n）的不稳定上同调。这个项目的第二个目标，与R. Charney合作，是为直角Artin群（一个既包括自由群又包括自由阿贝尔群的类）开发一个外太空的模拟。第三部分是由PI先前关于Out（F_n）和Aut（F_n）的同调稳定性的工作所激发的。为证明这些稳定性结果而开发的技术适用于与低维拓扑相关的其他群序列，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）刚性特性的研究。一个贯穿数学和科学的现象是，通过将复杂结构所包含的信息以图形和树的形式编码，通常可以更容易地理解复杂结构。这方面的一个例子是使用图和树来描述一个称为群的代数对象可能分裂成更简单的部分。当存在许多可能的分裂时，可以构造形变空间来度量歧义。在这个项目中，我们通过考虑群的对称性对这种变形空间的作用来研究群。

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.