Graphs, Trees and Geometric Group Theory

本课题研究相关度量对象的自同构群和变形空间。激励实例包括作用于平面n环面变形空间的GL（n，Z）群和作用于基本群Fn的紧度量图变形空间的Out（Fn）群。该项目有四个主要组成部分，从这些基本的例子出发，向不同的方向发展。第一个部分，与Martin Bridson合作，研究了Out（Fn）的刚性性质，限制了Out（Fn）和Out（Fm）之间映射的可能性，并限制了Out（Fn）在球体、可收缩流形和CAT(0)空间上的可能作用。第二个部分，与Ruth Charney合作，针对直角Artin群的自同构群。在最近的工作中，Charney和PI证明了Out（Fn）和GL（n，Z）所共有的许多性质实际上是所有直角Artin群的外部自同构群所共有的。使用的工具主要是代数工具，但本项目将使用CAT(0)几何开发新的几何工具。第三个分量，与John Smillie联合，考虑任意属平面的变形空间。该方案是用固定数量的奇异点定义标记平移曲面空间的边界化，该边界化下降到这些平移曲面的模空间的紧化。这是由SL（n， Z）和Out（Fn）空间的类似边界所激发的，并结合了两者的思想，对于研究这些群的上同调性质应该是有用的。该项目的第四部分，与Jim Conant和Martin Kassabov合作，回到Out(Fn)，并通过Kontsevich发现的Out（Fn）的上同调与与李操作相关的某个李代数的上同调之间的联系来研究Out（Fn）的理性上同调。数学中一个强大的工具是用代数形式对几何对象的结构进行编码。人们可以用代数思想来研究几何对象，或者用几何思想来研究代数对象。这个建议将这一思想引入到下一个层次：人们可以将代数对象的{\it变换}组或{\it自同构}与相关几何对象的{\it变形}空间联系起来。潜在的代数对象非常简单：它们是自由群、自由阿贝尔群和直角Artin群，但它们的自同构群非常复杂，仍然知之甚少。类似地，相关的代数对象（树、欧几里得空间和CAT(0)空间）并不复杂，但它们的变形空间表现出复杂的行为，这在纯数学和应用数学的许多领域都有影响。该项目将使用拓扑和几何工具来理解这些变形空间，并将获得的信息转化为关于自同构群的新信息。

This project studies automorphism groups and deformation spaces of related metric objects. The motivating examples include the group GL(n,Z) acting on the deformation space of flat n-tori and the group Out(Fn) acting on the deformation space of compact metric graphs with fundamental group Fn. The project has four main components, which take off from these basic examples in various directions. The first component, joint with Martin Bridson, studies rigidity properties of Out(Fn) which limit the possibilities for maps between Out(Fn) and Out(Fm) and constrain the possible actions of Out(Fn) on spheres, contractible manifolds and CAT(0) spaces. The second component, joint with Ruth Charney, targets automorphism groups of right-angled Artin groups. In recent work Charney and the PI have shown that many properties shared by Out(Fn) and GL(n,Z) are in fact shared by the outer automorphism groups of all right-angled Artin groups. The tools used have been largely algebraic, but this project will develop new geometric tools using CAT(0) geometry. The third component, joint with John Smillie, considers deformation spaces of flat surfaces of arbitrary genus. The proposal is to define a bordification of the space of marked translation surfaces with a fixed number of singular points which descends to a compactification of the moduli space of such translation surfaces. This is motivated by analogous bordifications of spaces for SL(n, Z) and Out(Fn) and combines ideas from both, and should be useful in studying cohomological properties of these groups. The fourth part of the project, joint with Jim Conant and Martin Kassabov, returns to Out(Fn) and investigates the rational cohomology of Out(Fn) via the connection found by Kontsevich between this cohomology and the cohomology of a certain Lie algebra associated to the Lie operad.A powerful tool in mathematics is to encode the structure of a geometric object in an algebraic form. One can then use algebraic ideas to study the geometric object, or geometric ideas to study the algebraic object. This proposal uses a bootstrap of this idea to the next level: one can relate the group of {\it transformations}, or {\it automorphisms} of an algebraic object to the space of {\it deformations} of an associated geometric object. The underlying algebraic objects are quite simple: they are free groups, free abelian groups and right-angled Artin groups, but their automorphism groups are remarkably complex and still poorly understood. Similarly, the associated algebraic objects (trees, Euclidean spaces and CAT(0) spaces) are uncomplicated but their deformation spaces exhibit complex behavior, which has implications in many areas of pure and applied mathematics. The project will employ topological and geometric tools to understand these deformation spaces and translate the information obtained into new information about automorphism groups.