Graphs, Trees and Geometric Group Theory

本项目研究相关度量对象的自同构群和形变空间。激发的例子包括群GL（n，Z）作用在平坦n-环面的变形空间上和群Out（Fn）作用在具有基本群Fn的紧度量图的变形空间上。该项目有四个主要组成部分，它们从这些基本示例出发，向各个方向发展。第一部分与Martin Bridson合作，研究Out（Fn）的刚性性质，这些性质限制了Out（Fn）和Out（Fm）之间映射的可能性，并限制了Out（Fn）在球面、可收缩流形和CAT（0）空间上的可能作用。第二个组成部分，联合露丝查尼，目标自同构群的直角阿廷组。在最近的工作中，Charney和PI证明了Out（Fn）和GL（n，Z）的许多共同性质实际上是所有直角Artin群的外自同构群所共有的。所使用的工具主要是代数的，但本项目将使用CAT（0）几何开发新的几何工具。 第三个组成部分，与约翰Smillie联合，考虑任意亏格的平坦曲面的变形空间。该建议是定义一个bordification的空间标记的翻译表面与一个固定数量的奇点下降到一个紧化的模空间的这种翻译表面。这是由SL（n，Z）和Out（Fn）的空间的类似边界化所激发的，并结合了两者的思想，并且应该在研究这些群的上同调性质时有用。 该项目的第四部分，与吉姆·科南特和马丁·卡萨博夫联合， 回到Out（Fn），通过Kontsevich发现的Out（Fn）的有理上同调和与李运算相关的李代数的上同调之间的联系，研究Out（Fn）的有理上同调。 然后，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。 这个提议将这个想法引导到下一个层次：人们可以将代数对象的{\it transformations}或{\it automorphisms}的群与相关几何对象的{\it deformations}的空间联系起来。 基本的代数对象非常简单：它们是自由群、自由阿贝尔群和直角阿廷群，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，这在许多纯数学和应用数学领域都有意义。 该项目将雇用 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.