Finite Decomposition Complexity Of Thompson Group

In this thesis, we study some problems in coarse geometry. There are many funda-mental conjectures in this area such as coarse Baum-Connes conjecture, coarse Novikov conjecture and Borel conjecture. Guentner-Tessera-Yu introduced a concept called finite decomposition complexity for metric spaces and proved that a bounded geometry metric space having finite decomposition complexity has Property A. Therefore, coarse Baum-Connes conjecture is true for a bounded geometry metric space having finite decomposition complexity. For a finitely generated group G (considered as a metric space when equipped with word-metric), if G has finite decomposition complexity, then coarse Baum-Connes conjecture is true for G.On the other hand, Thompson's group F is an attractive object in geometric group theory. It is a long-standing open problem to determine whether F is amenable. So far, we still haven't solved the problem of the amenability of F. Rufus Willett has proved that every amenable group has Property A in [69]. Thus the question about whether F has Property A or not has wide significance. Besides, a finitely generated group having finite decomposition complexity has Property A. So the question about the finite decomposition complexity of F is interesting. It might give us a method to solve the problem of the amenability of F.This paper contributes to the study of the geometric properties of Thompson's group F and some fundamental properties of finite decomposition complexity. At first, questions concerning distortion functions of subgroups in a finitely generated group is a topical sub-ject in geometric group theory. In the paper, by using the important tools of the reduced forest diagrams and the reduced tree diagrams, we prove that the wreath products F?Z and Z?Z are quasi-isometrically embedded subgroups of Thompson's group F. Then we focus on the finite decomposition complexity of some concrete metric spaces. For example, the subgroups of Thompson's group F. To make the property of finite decom-position complexity quantitative, a countable ordinal "the complexity" can be defined for a metric space with finite decomposition complexity. We obtained "the complexity" of the subgroup Z?Z of Thompson's group F, i.e., Z?Z?D?, where?is the smallest infinite ordinal number. Meanwhile, we proved that Thompson's group F equipped with the word-metric with respect to the infinite generating set{x0.x1,...xn,...} does not have finite decomposition complexity. At last, we showed that there is a injective Lips-chitz map from Thompson's group F equipped with the word-metric with respect to the finite generating set{x0,x1}to the linear group H equipped with a metric induced by a pseudo-length function. However, this map is not. proper.