Thompson群F的有限分解复杂度和几何性质

The project contributes to the study of Thompson''s group F,equipped with the word-metric with respect to the finite generating set. We are going to study the geometric properties of F by using the important tools of the reduced tree diagrams and the reduced forest diagrams. The main contents include the following aspects：.1. find a simple and intuitive formula of the word-metric of F with respect to the finite generating set. 2. study the exact complexity of the subgroups Z?Z,(Z?Z)?Z,((Z?Z)?Z）?Z …of F. 3. Focus on the properties of the F?lner functions and F?lner sets of F. We want to describe the problem of amenability of F by using its F?lner functions and F?lner sets. The project is planned to solve the problem of finite decomposition complexity of Thompson''s group F which is most likely to provide us a way to determine whether F is amenable or F has Property A.

本项目的主要研究对象是Thompson群F赋予有限生成集的词度量后形成的度量空间。期望利用研究Thompson群F的两个重要工具-约化树图和约化森林图来研究它的几何性质。主要研究内容有以下几个方面：1、找到Thompson群F关于有限生成集的一个简单、直观的词度量公式；2、研究F的圈积子群序列Z?Z,(Z?Z)?Z,((Z?Z)?Z）?Z …的精确复杂度；3、研究Thompson群F的F?lner函数和F?lner集合的性质，利用F的F?lner函数和F?lner集合刻画F的顺从性问题；本项目拟解决Thompson群F的有限分解复杂度问题，这极有可能给我们提供一种方法来解决Thompson群F是否具有顺从性或具有性质A等公开问题。

本项目找到了任何有限生成群H和整数群Z的圈积群HwrZ的一个词度量公式。同时利用约化树图和约化森林图，证明了Thompson群F的圈积子群FwrZ和ZwrZ都是F的拟等距嵌入的子群。另外还证明了具有性质SRD的可数离散群上的速降函数全体组成的空间是约化Banach代数谱不变的稠密Frechet子代数，双曲群具有性质SRD。