Asymptotic invariants of groups and subgroups

本文由三个部分组成，分别讨论了相对双曲群、它们的推广和剩余有限群的渐近不变量。这个建议的第一部分是受到一些重要的纯代数问题的启发，这些问题可以用PI发展的相对双曲群的几何小抵消理论来解决。第二部分提出了相对双曲性的一个推广，它包含了许多有趣的群在双曲空间上“很好”地作用（如映射类群、自由群的外自同构群、群图的基本群等）。第三部分的建议受到关于剩余有限群的三个渐近不变量的重要开放问题的启发：秩梯度、L2-betti数和群作用的代价。这个项目的总体思路是发展新的方法来研究几何和群与子群的渐近不变量。所提出的方法主要是基于相对双曲群理论的最新进展。这个项目的许多动机来自数学的其他分支，如非交换几何、低维拓扑和拓扑动力学。如果成功，该项目将有助于在几何群论中创造新的强大工具。另一方面，提出的问题的解决可能会对数学的其他领域产生强烈的影响。

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.