Boundaries of Hyperbolic and Relatively Hyperbolic Groups

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The concept of a group appears in several areas of pure and applied mathematics. Often a group has a topological space, its "boundary at infinity", that gives us some information about the group. The study of this brings the fields of topology, algebra, and dynamics into play. This project addresses the broad question of how much the boundary of a group determines the group. The principal investigator continually seeks to spread mathematical knowledge to a wide audience, including under-represented groups in mathematics and people working in industry and technology. This is done through speaking at and organizing conferences in a wide variety of venues aimed at wide audiences, and encouraging people in these groups (including industry) to attend her classes. In addition, all of the PI's mathematical results and findings are freely available to the public. The project addresses important and fundamental problems involving which hyperbolic or relatively hyperbolic groups can have certain boundaries. For one part of the project the principal investigator plans to extend known results about hyperbolic groups with planar or two-sphere boundaries to the relatively hyperbolic case, strengthening the connection with Kleinian groups and the fundamental groups of 3-manifolds. The proof methods sometimes are similar to the case of hyperbolic groups but often require different techniques. For the other part of the project she is exploring the wealth of planar boundaries than can occur as the complement of round disks in the 2-sphere.