Hyperbolic geometry in group theory

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The proposed project consists of two parts. The central theme of the first part is a conjecture, which can roughly be stated as follows: the class of groups admitting a non-elementary and "reasonably proper" action on a hyperbolic space is the same for any reasonable interpretation of the phrase "reasonably proper"; moreover, it coincides with the class of groups with non-degenerate hyperbolically embedded subgroups introduced by Dahmani, Guirardel, and the PI. The second part of the proposal is devoted to a more detailed study of groups with hyperbolically embedded subgroups. In particular, the PI plans to study questions related to rigidity, small cancellation theory, and applications of the general theory to groups acting on trees, one-relator groups, graph products, and fundamental groups of 3-manifolds.The general goal of the project is to better understand the scope and limitations of geometric methods based on negative curvature in group theory. As a first step towards this goal, the PI proposes to analyze various negative curvature phenomena of algebraic, geometric, dynamical, and analytic nature occurring in different branches of group theory. This analysis is expected to lead to new general results with possible applications in group theory, 3-dimensional geometry, the theory of operator algebras, and other branches of mathematics.