Geometry and dynamics of outer automorphism groups of free groups

几何群论是利用代数、拓扑学和分析等数学工具来研究几何对象及其对称性的群，特别强调“大尺度”或“准等距”工具，这些工具用于研究几何对象，就像通过越来越强大的望远镜观察它一样。在这个项目中，PI将应用几何群论的方法来研究拓扑学家所称的有限网络的隐藏对称性，即“图”。这些隐对称群被称为“自由群的外自同构群”。这个项目的一个成果将是解决自由群的外自同构群的大规模几何中的重要问题，并通过这些解决方案产生新的大规模几何工具，供几何群论社区广泛使用。另一个结果将是建立一个新的工具来研究自由群的外自同构群的动力学性质。有限秩自由群的外自同构群，记为(F_N)，在1930年由Nielsen和Whitehead首先研究了S。近几十年来，随着诸如卡勒-沃格特曼外层空间和Bestvina-Feighn-Handel的相对轨道图等几何工具的发展，人们对OUT(F_N)的兴趣迅速扩大。最近，Out(F_N)正在用准等距几何进行研究，在这一努力中，首席研究员发挥了主导作用。在这个项目中，PI将研究Out(F_N)的有限生成子群在双曲度量空间上的作用，试图为Out(F_N)建立一个类似于Masur和Minsky在1990年‘S建立的映射曲面类群的层次理论。作为应用和建立有用的层次理论的指导，PI计划研究和解决Out(F_N)的大规模几何中的各种问题，包括：Out(F_N)的有限生成子群的第二有界上同调的计算；以及Out(F_N)的拟等距刚性的早期研究。PI还计划对某些Out(F_N)不变动力系统进行早期研究，这些动力系统类似于曲面及其不变集的Teichmuller测地流。

Geometry group theory is the study of geometric objects and their groups of symmetries using mathematical tools such as algebra, topology, and analysis, with a special emphasis on "large scale" or "quasi-isometric" tools which are used in studies of a geometric object as if viewing it through more and more powerful telescopes. In this project the PI will apply the methods of geometric group theory to the study of hidden symmetries of finite networks known to topologists as "graphs". These groups of hidden symmetries are known as "outer automorphism groups of free groups". One outcome of this project will be to solve important problems in the large scale geometry of outer automorphism groups of free groups and through these solutions to produce new tools of large scale geometry for broad use by the geometric group theory community. Another outcome will be to build tools for a new study of the dynamical properties of outer automorphism groups of free groups.Outer automorphism groups of finite rank free groups, denoted Out(F_n), were first examined in the 1930's by Nielsen and Whitehead. Interest in Out(F_n) has expanded rapidly in recent decades with the development of geometric tools such as: the Culler-Vogtmann outer space; and the relative train track maps of Bestvina-Feighn-Handel. More recently Out(F_n) is being studied using quasi-isometric geometry, an effort in which the principal investigator has played a leading role. In this project the PI will study actions of finitely generated subgroups of Out(F_n) on hyperbolic metric spaces, attempting to build for Out(F_n) an analogue of the hierarchy theory for mapping class groups of surfaces that was built in the 1990's by Masur and Minsky. As applications, and as guides to building a useful hierarchy theory, the PI plans to study and resolve various problems in the large scale geometry of Out(F_n) including: computation of the 2nd bounded cohomology of finitely generated subgroups of Out(F_n); and early investigations into quasi-isometric rigidity for Out(F_n). The PI also plans to carry out early investigations of certain Out(F_n)-invariant dynamical systems that are analogous to the Teichmuller geodesic flow of a surface and its invariant sets.