Homeomorphism Groups of One-manifolds: Rigidity and Regularity

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One dimensional manifolds are fundamental objects in mathematics, and their structure and properties have attracted the attention of mathematicians for a long time. The focus of the project is the structure of the group of continuous symmetries of one-manifolds, including the local structure of the group of symmetries and the effect of smoothness on algebraic properties. The project will resolve several basic open questions about the structure of these groups and investigate new algebraic phenomena. The ideas and methods in the project will be broadly applicable to other fields, and will lie at the interface of topology, geometric group theory, and dynamical systems.The structure of the homeomorphism groups of one-manifolds, especially compact one-manifolds, has received much attention over the years. Recently, this field has experienced tremendous development. The project will contribute to these advances by studying the local topology of the space of representations of a closed surface group into homeomorphisms of the circle, the local connectivity of which is a long standing open problem. Moreover, the project addresses the effect of regularity on the algebraic structure of finitely generated subgroups of homeomorphism groups on compact manifolds, and will construct groups which can act with prescribed levels of regularity but which are not smoothable to higher levels of regularity. The regularity of actions of Thompson's group F will also be addressed, with the goal of establishing the Brin-Sapir conjecture for smooth actions of F on compact manifolds.