Coarse Geometric Properties Of The R-tree

In this thesis, we discuss some properties of R-trees under the ideas of coarse ge-ometry. We extend Property A to general measure metric spaces, and construct a mea-sure on any R-tree to make it become a measure metric space. We prove that, as measure metric spaces,R-trees have extended property A. We give an example of group which does not have finite decomposition complexity, obtain the results that R-trees have finite decomposition complexity and finite presented groups that can act isometric and freely on R-trees have finite decomposition complexity.