Groups acting on hyperbolic spaces

Vanderbilt University, Nashville TN

Investigators

Abstract

Award: DMS 1612473, Principal Investigator: Denis Osin Geometric group theory studies algebraic objects (groups) by visualizing them as sets of transformations of geometric objects (metric spaces). In the 1980s, Gromov revolutionized the field by introducing the notion of a hyperbolic space and outlining a broad program of study of isometry groups of such spaces. Intensive work in this direction has resulted in the rich theory of hyperbolic and relatively hyperbolic groups. A further generalization, the class of acylindrically hyperbolic groups, was recently suggested by the Principal investigator; it received considerable attention in the papers of the PI and others over the past few years. The main goal of the proposed project is to continue this work and to make further advances in the study of groups acting on hyperbolic spaces. More specifically, the proposed project consists of 4 parts. The main objective of the first part is to better understand the relation between various manifestations of negative curvature in geometry, analysis, and topological dynamics. The second part is devoted to the study of group theoretic Dehn surgery, an algebraic generalization of Thurston's theory of hyperbolic Dehn filling introduced in earlier papers of the PI. The PI suggests some further directions with potential applications to the study of group von Neumann algebras. The third part is focused on applications of geometric methods to the study of permutation groups. In particular, the PI proposes a way of solving several long standing open problems about factorizable groups. In the last part, the PI defines the poset of actions of a given group on hyperbolic spaces and proposes several natural questions about it. One direction of particular interest here is the study of various rigidity phenomena analogous to the marked length spectrum rigidity of hyperbolic manifolds.