Relative hyperbolicity and asymptotic invariants of groups

Vanderbilt University, Nashville TN

Investigators

Abstract

This project is devoted to the study of relatively hyperbolic groups and their applications in low--dimensional topology. It consists of three main parts. First part deals with small cancellation theory over relatively hyperbolic groups. Namely we intend to use methods elaborated by the PI to construct new groups with 'exotic' properties that solve some long--standing problems in the classical theory of groups. The second part is motivated by the Virtual Haken Conjecture and is based on a generalization of Thurston's hyperbolic Dehn surgery theory in the context of relatively hyperbolic groups. The third part concerns new results about group embeddings. The ultimate goal here is to obtain a relatively hyperbolic version of the Higman embedding theorem. Our approach to these problems is based on a new methods related to asymptotic and combinatorial analysis of van Kampen diagrams. The notion of a relatively hyperbolic group was first proposed by Gromov in 1987 to generalize many examples of algebraic and geometric nature. Since then the theory has been elaborated from different points of views and now it is one of the most prominent parts of the group theory. The class of relatively hyperbolic groups encompasses many examples of interest such as fundamental groups of finite volume hyperbolic manifolds, geometrically finite convergence groups, mapping class groups, fully residually free groups, etc. Rather than just generalizing the theory of hyperbolic groups to the relative case, this project is aimed at the study of new and sometimes unexpected phenomena which are invisible in the context of ordinary hyperbolicity. The close interaction between the theory of relatively hyperbolic groups and geometry, low-dimensional topology, the theory of dynamical systems, provides a rich source of problems and motivations for our project.