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非常重要。 这两类群体之间的类比常常推动研究。首席研究员将与雷曼学院的 Michael Handel 合作，研究与 MCG(S) 类比的几个关于 Out(F_n) 的问题。他将研究 Out(F_n) 子群的分类，该分类类似于 Ivanov 的 MCG(S) 子群分类。他将研究与 F_n 相关的复形几何，目标是找到关于 S 曲线复形几何的马苏尔-明斯基定理的类似物。这两项研究可能导致对 Out(F_n) 有界上同调的研究，类似于 Bestvina-Fujiwara 关于 MCG(S) 有界上同调的结果。他将寻找推广伪阿诺索夫表面同胚的外自同构的同胚代表。在单独的工作中，首席研究员将与 Jason Behrstock、Bruce Kleiner 和 Yair Minsky 合作研究映射类组的准等距刚性。他还将研究细化弱相对双曲性的映射类群的几何性质。群论是对对称性抽象性质的研究。几何群理论更紧密地关注几何对象的对称群，通过将抽象给定群实现为适当几何对象的对称群，它可以用来极大地丰富我们关于抽象给定群的知识。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，这些群自然地产生于拓扑考虑，但本质上不是几何的：映射类群产生于二维表面的拓扑对称性；自由群的外自同构群由一维图的同伦对称产生。首席研究员将研究几何对象，其对称群是通过映射曲面的类群或自由群的外自同构群来实现的，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.