Geometry of Mapping Class Groups and Outer Automorphism Groups

曲面MCG（S）的映射类群和自由群Out（F_n）的外自同构群在几何群论中是非常重要的。 这两类群体之间的类比往往推动研究。本文的主要研究者将与Lehman学院的Michael Handel合作，研究Out（F_n）的几个问题，这些问题是由与MCG（S）的类比所激发的。他将研究Out（F_n）的子群的分类，它类似于Ivanov对MCG（S）的子群的分类。他将研究与F_n有关的复形的几何，目标是找到关于S的曲线复形几何的Masur-Minsky定理的类似物。这两个研究可以导致Out（F_n）的有界上同调的研究，类似于Bestvina-Fujiwara关于MCG（S）的有界上同调的结果。他将寻找同胚代表外自同构推广伪Anosov表面同胚。在单独的工作中，与Jason Behrstock，布鲁斯Kleiner，和Yair Minsky，首席研究员将研究映射类群的拟等距刚性。他还将研究映射类群的几何性质，以改进弱相对双曲性。群论是对对称性的抽象性质的研究。几何群论更紧密地集中在几何对象的对称群上，它可以用来极大地丰富我们对抽象给定群的知识，通过将其实现为适当几何对象的对称群。两个非常重要的类的群体，研究以这种方式是映射类群体的表面和外部自同构群的自由群体。在这两种情况下，这些群自然地产生于本质上不是几何的拓扑考虑：映射类群产生于二维曲面的拓扑对称性;自由群的外自同构群产生于一维图的同伦对称性。主要研究人员将继续研究几何对象的对称群实现映射类组的表面或外部自同构群的自由群体，与应用程序的结构，这些群体。

Mapping class groups of surfaces MCG(S) and outer automorphism groups of free groups Out(F_n) are very important in geometric group theory. Analogies between these two classes of groups often drive research. The principal investigator, working jointly with Michael Handel of Lehman College, will study several problems about Out(F_n) that are motivated by analogies with MCG(S). He shall investigate a classification of subgroups of Out(F_n) which is analogous to Ivanov's classification of subgroups of MCG(S). He shall investigate the geometry of complexes related to F_n, with the goal of finding analogues of the Masur-Minsky theorems about the geometry of the curve complex of S. These two investigations could lead to a study of the bounded cohomology of Out(F_n), in analogy to results of Bestvina-Fujiwara about bounded cohomology of MCG(S). He shall search for homeomorphic representatives of outer automorphisms that generalize pseudo-Anosov surface homeomorphisms. In separate work, joint with Jason Behrstock, Bruce Kleiner, and Yair Minsky, the principal investigator will study quasi-isometric rigidity of mapping class groups. He shall also study geometric properties of mapping class groups that refine weak relative hyperbolicity.Group theory is the study of abstract properties of symmetry. Geometric group theory focusses more tightly on symmetry groups of geometric objects, and it can be used to greatly enrich our knowledge about an abstractly given group, by realizing it as the group of symmetries of an appropriate geometric object. Two very important classes of groups that are studied in this manner are mapping class groups of surfaces and outer automorphism groups of free groups. In both cases, these groups arise naturally from topological considerations that are not, at their heart, geometric in nature: mapping class groups arise from the topological symmetries of 2-dimensional surfaces; outer automorphism groups of free groups arise from the homotopic symmetries of 1-dimensional graphs. The principal investigator will pursue the study of geometric objects whose symmetry groups are realized by mapping class groups of surfaces or by outer automorphism groups of free groups, with applications to the structure of these groups.