Geometry of the outer automorphism group of a free group

主要研究者（与纽约市立大学Lehman学院的Michael Handel合作）正在研究秩为n的自由群的外自同构群和外空间Out（F_n）和X_n的渐近性质和代数性质。 研究的方法是发展Masur和Minsky的层次理论的一个类似物，用来研究曲面S的映射类群MCG（S）。基于他最近与Handel关于Out（F_n）的各种自然子群的畸变和非畸变的发现，PI目前正致力于使用自由群F_n的自由分裂作为曲面上本质曲线系的自然模拟，以及自由分裂的“精化复形”作为次表面曲线复形的自然模拟（至少在某些情况下）。特别地，PI正在研究如何使用定义在X_n的脊上的投影映射，其目标是精化复合体和密切相关的对象。利用这些投影图，PI研究的中心焦点是为Masur-Minsky和的Out（F_n）建立一个适当的模拟，这是MCG（S）中一个非常有用的词度量公式。 潜在的数学应用包括提高对Out（F_n）共轭问题计算复杂性的理解。PI还继续与Handel合作开展一个长期的联合项目，研究Out（F_n）的子群分类理论。在一个单独的项目中（与纽约市立大学莱曼学院的Jason Behrstock合作），PI致力于将层次理论应用于MCG（S），以精确理解MCG（S）共轭问题的计算复杂性。几何群论是对几何对象的对称群的研究。在某些情况下，几何对象早在其对称群之前就被理解了，例如，平面在古代就被知道了，但它的对称性--它的刚性运动--直到世纪才被完全理解。在其他情况下，抽象群在发现具有该对称群的几何之前就已经知道了。 主要研究者研究了一个抽象群，称为“秩为n的自由群的外自同构群”，记为Out（F_n），它是在世纪上半叶发现的。它的相关几何对象，“外层空间的秩n”表示为X_n，只是在20世纪80年代由数学家马克·卡勒和卡伦·沃特曼发现。空间X_n是理解有限网络的一个重要对象，它可以被看作是一个包含一系列复杂网络的单一包，这些网络被称为“标记图”，它们可以通过扭曲一个非常简单的网络（“数学家的玫瑰”）来获得，即在一个点上连接的n个环的集合。继数学家Michael Gromov在20世纪70年代后期提出的几何群论的广泛研究计划之后，PI对Out（F_n）的研究与X_n的渐近行为的研究紧密交织在一起，这意味着它的行为非常非常远离固定的观点。PI对Out（F_n）和X_n的渐近分析涉及比较标记图的有效与无效失真。目前PI研究的中心焦点是发现一个测量标记图有效失真的通用公式。

The Principal Investigator (jointly with Michael Handel of Lehman College,CUNY) is studying asymptotic and algebraic properties of Out(F_n) and X_n, the outer automorphism group and outer space of a free group of rank n. The method of study is to develop an analogue of the hierarchy theory of Masur and Minsky used to study MCG(S), the mapping class group of a surface S. Based on his recent discoveries with Handel of distortion and nondistortion of various natural subgroups of Out(F_n), the PI is currently focussed on using free splittings of the free group F_n as a natural analogue of essential curve systems on surfaces, and on the "refinement complex" of a free splitting as a natural analogue (at least in certain cases) for the curve complex of a subsurface. In particular, the PI is studying how to use projection maps defined on the spine of X_n, whose targets are refinement complexes and closely related objects. Using these projection maps, the central focus of the PI's research at the moment is to formulate an appropriate analogue for Out(F_n) of the Masur--Minsky sum, a very useful formula for the word metric in MCG(S). Potential mathematical applications include an improved understanding of the computational complexity of the conjugacy problem for Out(F_n). The PI also continues to work on a longer term joint project with Handel on subgroup classification theory for Out(F_n). In a separate project (joint with Jason Behrstock of Lehman College, CUNY) the PI works on applying hierarchy theory for MCG(S) to obtain a precise understanding of the computational complexity of the conjugacy problem for MCG(S).Geometric group theory is the study of symmetry groups of geometric objects. In some cases the geometric object was understood long before its symmetry group, for instance the flat plane was known in antiquity, but its symmetries --- its rigid motions --- were fully understood only in the 19th century. In other cases, the abstract group was known before the discovery of a geometry with that symmetry group. The Principal Investigator studies an abstract group called the "outer automorphism group of a free group of rank n", denoted Out(F_n), which was discovered in the first half of the 20th century. Its associated geometric object, the "outer space of rank n" denoted X_n, was discovered only in the 1980's by the mathematicians Marc Culler and Karen Vogtmann. The space X_n, a key object for understanding finite networks, may be regarded as a single package encompassing a complicated array of networks known as "marked graphs" that can be obtained by distorting one very simple network, the "mathematician's rose", a collection of n loops conjoined at a single point. Following a broad research program for geometric group theory that was laid out in the late 1970's by the mathematician Michael Gromov, the PI's study of Out(F_n) is closely intertwined with a study of the asymptotic behavior of X_n, meaning its behaviour very very far away from a fixed point of view. The PI's asymptotic analysis of Out(F_n) and X_n involves comparing efficient versus inefficient distortions of marked graphs. The central focus of the PI's research at the moment is to discover a general formula for measuring efficient distortion of marked graphs.