Mapping Class Group and Fiber Bundles

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This project is concerned with the study of spaces (manifolds), especially in dimensions 2, 3, and 4. In the study of these objects, a fundamental role is played by the mapping class group, which captures the symmetries of a space and encodes gluing data for building new spaces, such as fiber bundles. The study of mapping class groups is a meeting ground for many areas including geometric group theory, dynamics, homotopy theory, algebraic geometry, and mathematical physics. This project focuses on using mapping class groups to understand geometric properties of fiber bundles. The project provides research training opportunities for graduate students and includes organizing the Brown-Yale Geometry and Topology Conference. The project has three main objectives. (1) Find new examples of non-flat bundles, especially surface bundles whose base is low-dimensional. Research in this direction will increase our understanding of the dynamics of surface homeomorphisms, the algebra of mapping class groups, and their relation. (2) Establish new connections between the dynamical properties of a pseudo-Anosov surface homeomorphism and the (non)arithmeticity of the associated hyperbolic mapping torus, thereby building on work of Thurston and Bowditch-Maclachlan-Reid. (3) Provide new examples of convex-cocompact subgroups of the mapping class group. This work is aimed at a conjecture of Farb-Mosher regarding purely pseudo-Anosov subgroups of mapping class groups, and it also has connections to 3-manifold topology. The project will also support the advising of students and the organization of a conference.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.