Boundaries of Groups

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A central theme in mathematics for centuries has been the interaction between algebra and geometry. The usual direction is to study the set of symmetries of a geometric object of interest. In geometric group theory, this becomes a two-way street in that algebraic objects (such as groups) are considered as geometric objects in their own right. Hyperbolic geometry is a subject going back to work of Bolyai, Gauss and others in the 19th Century, but it also plays a central role in modern geometry, due to the influence of Thurston and Gromov. This project centers around a central question in geometric group theory, the Cannon Conjecture, about the difference (in three dimensions) between classical hyperbolic geometry and the coarse notion due to Gromov, in the presence of a large group of symmetries. Broader impacts of this project include research training opportunities for graduate students.Over the last decade the principal investigator, along with Manning and others, has developed many tools involving relatively hyperbolic Dehn filling, which gives strong control on certain kinds of quotients of relatively hyperbolic groups. This project leverages this work to study hyperbolic and relatively hyperbolic groups whose boundary at infinity is a two-sphere. The Cannon Conjecture predicts that such groups are virtually Kleinian groups. This project proposes various approaches to this and related conjectures. With Haissinsky, Manning, Osajda, Sisto and Walsh, the PI continues to develop a theory of drilling hyperbolic groups with two-sphere boundary. The PI will investigate possible quasi-isometries between hyperbolic and relatively hyperbolic groups with 2-sphere boundaries. With Wilton, the PI will develop a notion of coarse sectional curvature, with applications to coherence and local quasi-convexity of certain hyperbolic groups. In a different but related direction, with Einstein the PI will continue to study relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes.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.