Asymptotic invariants of groups and subgroups

Vanderbilt University, Nashville TN

Investigators

Abstract

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.