Virtual properties of 3-manifold groups

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Under the grant, the PI will study topology of 3-manifolds in terms their fundamental groups. The research will be based on recent progress in the Virtual Fibration Conjecture and the Virtual Haken Conjecture. Ingredients about surface subsubgroups in 3-manifolds will be summarized and generalized, and will be applied to approach existing problems including the virtual positivity of representation volumes and the linearity of 3-manifold groups. The project will also study the virtual properties of surface automorphisms. This is closely related to the surface bundle case of Virtual Homological Torsion Conjecture. The PI proposes to apply methods from geometric group theory, dynamical systems, representation theory, and low dimension topology.Spaces and transformations provide an important perspective on mathematically modelling our world. Natural sciences and certain social sciences utilize them as abstract tools to understand symmetry of various systems. The PI will study geometric transformations on 3-dimensional spaces, which form an essential family for many applications involved in physics, computer sciences, and other disciplines. This family of transformations have been proved to be powerful and mathematically comprehensible, thanks to recent solutions of long existing conjectures. The principal pursuit of this project is to attack several existing problems in low dimensional topology with the new techniques. Besides, the project endeavors to theoretically summarize recent development of the area, making the new ideas more accessible to working scientists from all disciplines.