Geometry of Free Group Automorphisms: Beyond the Frontier

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Group is a fundamental concept in many areas of modern Mathematics. The project will continue the study of dynamics and geometry of the automorphism group of a free group, a key object in modern mathematics exhibiting a rich variety of new and unexplored phenomena. The project will capitalize on the recent progress in the area. It also aims to open up new directions of research, such as a systematic study of endomorphisms of free groups. The PI will continue his work on preparing the next generation of STEM graduate and undergraduate students, through running an innovative blended seminar "Geometry, Groups and Dynamics", supervising vertically integrated undergraduate research projects in the Illinois Geometry Lab, directing PhD thesis research in mathematics, and so on. The PI will also continue his active participation in Wikipedia editing on mathematical topics, aiming to reach broad segments of general public and of other sciences and disciplines, and to educate them about the current state of mathematical research. The project will continue developing a free group counterpart of the "fibered face" theory for 3-manifolds and the study of mapping tori of automorphisms and endomorphisms of free groups. A key tool used in this study is the folded mapping torus associated to a train track map, together with a semi-flow, which is then studied as a dynamical system and replaces a fibered 3-manifold in the free group setting. Specific questions to be investigated include refining the understanding of the Bieri-Neumann-Strebel (BNS) invariant for the mapping tori groups, constructing a canonical version of the McMullen polynomial for them, investigating stable trees and Cannon-Thurston maps, and developing a systematic train track theory for free group endomorphisms.