Geometry of Mapping Class Groups and Outer Automorphism Groups

映射曲面类群MCG(S)和自由群OUT的外自同构群(F_N)在几何群论中是非常重要的。这两类群体之间的类比经常推动研究。主要研究人员将与雷曼学院的迈克尔·汉德尔合作，研究由与MCG(S)类比而引发的关于OUT(F_N)的几个问题。他将研究Out(F_N)的子群的分类，这类似于Ivanov对MCG的子群的分类(S)。他将研究与F_n有关的复形的几何，目的是找到类似于关于S的曲线复形几何的Masur-Minsky定理。这两个研究可以导致对Out(F_N)的有界上同调的研究，类似于Bestvina-Fujiwara(S)关于MCG的有界上同调的结果。他将寻找推广伪Anosov曲面同胚的外自同构的同胚表示。在与Jason Behrstock、Bruce Kleiner和Yair Minsky合作的单独工作中，首席研究员将研究映射类群的准等距刚性。他还将研究改进弱相对双曲性的映射类群的几何性质。群论是对对称的抽象性质的研究。几何群论更紧密地关注几何对象的对称群，它可以通过将抽象给定的群实现为适当几何对象的对称群来极大地丰富我们对它的认识。以这种方式研究的两类非常重要的群是曲面的映射类群和自由群的外自同构群。在这两种情况下，这些群都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2维曲面的拓扑对称；自由群的外部自同构群产生于1维图的同伦对称。主要的研究人员将继续研究其对称群是通过映射曲面的类群或通过自由群的外自同构群来实现的几何对象，并应用于这些群的结构。

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.