Geometry and Dynamics in Surfaces and Free Group Extensions

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The algebraic properties of the symmetries of an object are encoded by the mathematical structure known as a group. Formally, a group is simply a set equipped with a binary operation that satisfies two axioms formalizing the concept of undoing the operation. Mathematicians are concerned both with the overarching abstract algebraic structures that arise from these rules, and with concrete examples that have natural significance, such as the set of rigid motions of the plane, the configurations of a Rubik's cube, or the symmetries of a polyhedron. Here free groups, which have a particularly simple algebraic structure in which there is the minimal possible cancellation, play an essential role as building blocks from which many other groups are built. Free group extensions are a basic example of this sort of construction. Yet, despite being built from well-understood pieces, free group extensions remain rather mysterious. This project will investigate dynamical, algebraic, and geometric aspects of free group extensions and combine these perspectives to illuminate the broad structure of these groups. This expands on the investigator's prior work with collaborators exploring and relating the different ways a given group may be built as a free-by-cyclic group, as well as work on the coarse negative curvature of certain free group extensions.This project will focus on aspects of geometry, topology, algebra, and dynamics in the context of free group automorphisms and surface homeomorphisms. The investigator will continue his work with collaborators analyzing splittings of free-by-cyclic groups via the dynamics of certain semi-flows on 2-complexes. A main goal is to tie together all the monodromies of the different splittings via a uniformized action of the entire group on a tree and to use this to define canonical algebraic invariants and relate all the Cannon-Thurston maps to each other. In addition to strengthening these connections, the project will work to prove that the property of having a fully irreducible monodromy is shared by either all or none of the splittings of the group. Continuing his work with a collaborator, the investigator will study the geometry of general free group extensions and work to characterize Gromov hyperbolicity and establish rigidity results for this class of groups. With regard to the study of surface homeomorphisms, the investigator will work with collaborators to show that all small-dilatation pseudo-Anosov homeomorphisms have the simple structure predicted by the symmetry conjecture. Finally, the investigator will study large-scale geometric and dynamical properties of the Teichmuller space of a surface with an aim towards calculating expected thinness of triangles and lattice point asymptotics for natural sets of surface homeomorphisms.