Geometry of Mapping Class Groups and Outer Automorphism Groups

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.