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。如果只关注具有强几何性质的空间(如度量树或CAT(0)空间)，通常可以使G的作用集成为具有良好拓扑性质的拓扑空间本身，称为变形空间。G的自同构群通过扭曲每个动作来作用于变形空间，从而得到一个新的动作。在这个项目中，我们通过研究自同构群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.Culler定义的相关变形空间称为外层空间。最近，Morita利用Kontsevich发现的外层空间与某个无限维辛李代数之间的联系来检测OUT(F_N)的余圈，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，并计划用它们来研究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群开发一个类似外太空的东西，这类群既包括自由群，也包括自由阿贝尔群。第三个分量是由PI在Out(F_N)和Aut(F_N)的同调稳定性方面所做的工作得到的。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研究系统发育树的空间和调查OUT(F_N)的刚性性质。一个贯穿数学和科学的现象是，复杂的结构通常可以通过将它们包含的信息以图形和树的形式进行编码来更容易地被理解。例如，使用图和树来描述称为组的代数对象可能被分割成更简单的片段。当存在许多可能的分裂时，可以构造度量歧义的变形空间。在这个项目中，我们通过考虑群的对称性在这样的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.