The Coarse Property Of 3-manifold Group And Conditions For Coarse Embedding

In this thesis, we concerned on the problems in coarse geometry. There are many fundamental conjectures in this area such as coarse Baum-Connes conjecture, coarse Novikov conjecture and Borel conjecture. Guentner-Tessera-Yu introduced a property called finite decomposition complexity for metric spaces and proved the stable Borel conjecture holds for the aspherical manifolds whose fundamental groups have finite decomposition complexity [35]. So it is important to know which group has finite de-composition complexity. We proved the fundamental group of a compact 3-manifold has finite decomposition complexity provided the Thurston's Hyperboliable conjecture is true. So any group which can be realized as the fundamental group of a aspherical manifold of above kind satisfies the stable Borel conjecture.On the other side, K.Kasparov and G.Yu proved the Novikov conjecture with co-efficients holds for the metric spaces with bounded geometry, which admit coarse em-bedding into a uniformly convex Banach space [53]. we choose a special uniformly convex Banach space, (?), to study. We gave a sufficient and necessary condition for coarse embedding into (?). And we proved under some conditions, this property can be preserved under taking the union of metric space. As an application, we proved the ball under relative metric in relative hyperbolic group can be coarse embedded into (?) if and only if the subgroup is coarse embedded. We also discussed the embeddity of tree-graded spaces. We showed that the tree graded space with bounded∈-capacity admits a coarse embedding into uniformly convex Banach space if all its pieces admit coarse embedding into (?) uniformly.