Group Actions on Manifolds and Related Spaces: Regularity, Structure, and Complexity

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Manifolds are fundamental objects in mathematics which generalize and systematize space as we perceive it, and as such have been of great interests to mathematicians for a long time. One fruitful way in which manifolds can be studied is through their symmetries, which taken together form an algebraic object called a group. The focus of this project is the interaction between the structure of a manifold, both local (up close) and global (as an ensemble), and its symmetries, or group actions on manifolds for short. The project aims to resolve some basic questions in this area and investigate some new phenomena. Among the problems of particular interest are those concerning smoothness of group actions, in the sense of calculus. A resolution of some of these problems has the potential to yield insight into old open questions about exotic spheres, which are certain manifolds which can be deformed into standard spheres, but only in a highly singular way. Groups acting on manifolds are a broad class of objects which have received a large amount of attention, and their theory has developed rapidly in recent years. Planned broader impacts include research experiences for undergraduates, conference organization, and a book on right-angled Artin groups.This project will contribute to these advances in several ways. One area of focus for the project is critical regularity of group actions. Previous work with Kim will be extended in order to compute the critical regularity in one dimension of right-angled Artin groups, which would then be the first natural class of non-nilpotent groups whose precise one-dimensional critical regularity is both finite and known. Moreover, the principal investigator intends to develop critical regularity in higher dimensions, with the goal of encoding the diffeomorphism type of a compact manifold by a set of finitely generated groups. Such a result would give an algebraic approach to the 4-dimensional smooth Poincare conjecture. Another focus of the project is to develop a structure theory for thin subgroups of semisimple Lie groups, with the goal of resolving a conjecture of Shalom about the discreteness of their commensurators. The final main focus of the project is in the area of mapping class groups of surfaces, where the PI has outlined a program to prove the nonlinearity of mapping class groups, and to introduce logic into low dimensional topology by investigating the model theory of the curve graph of a surface.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.