This proposal consists of three parts devoted to relatively hyperbolic groups, their generalizations, and asymptotic invariants of residually finite groups. The first part of this proposal is inspired by some important purely algebraic problems, which can be solved using geometric small cancellation theory over relatively hyperbolic groups developed by the PI. The second part proposes a generalization of relative hyperbolicity, which encompasses many interesting groups acting "nicely" on hyperbolic spaces (e.g., mapping class groups, outer automorphism groups of free groups, fundamental groups of graphs of groups, etc.) The third part of the proposal is inspired by important open problems about three asymptotic invariants of residually finite groups: rank gradient, L2-betti numbers, and cost of group actions.
The general idea behind this project is to develop new methods to study geometry and asymptotic invariants of groups and subgroups. The proposed approach is mostly based on recent advances in the theory of relatively hyperbolic groups. Lots of motivation for this project come from other branches of mathematics such as non-commutative geometry, low-dimensional topology, and topological dynamics. If successful, the project will help to create new powerful tools in geometric group theory. On the other hand, solution the proposed problems will likely have strong impact on other areas of mathematics.
Intellectual merit. One of the main goals of the project was to develop a new approach to the study of groups acting on hyperbolic spaces. This is done in a long paper (joint with F. Dahmani and V. Guirardel), where we introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of peripheral structures of relative hyperbolicity groups, while the later one provides a natural framework for developing a geometric version of small cancellation theory. This approach was then used in other papers by the PI and his co-authors to study algebraic, geometric, and analytic properties of groups acting on hyperbolic spaces. Other results include the solution of a 15 years old problem about c-compact topological groups (Joint with A. Klyachko and A. Olshanskii) , complete classification of conjugacy growth functions of finitely generated groups (joint with PI's graduate student M. Hull), and a new quasi-isometric embedding theorem for groups, which was used to answers several open questions (joint with A. Olshanskii). Broader Impact. The PI actively involved undergraduate students, graduate students, and postdocs in his research. In 2010-2013, the PI taught several courses for graduate and advanced undergraduate students at Vanderbilt covering various topics in geometric group theory. these courses were partially based on the results of this and previous projects.
