Some Coarse Permanence Properties Of Metric Spaces

In this paper we study some permanent properties of metric spaces. Property generalized property A is the notion of metric geometry which implies the coarse Novikov conjecture .In the paper ,we shall prove the permanence property of generalized property A of metric spaces under large scale decompositions of finite depth.And we also study the strongly coarse embeddability into a uniformly convex Banach space. Under the act of coarse equivalent ,the strongly coarse embeddability has the permanent property.And finally ,we study the metric spaces which have finite decomposition complexity property and denote it FDG. namely, a finite generated group which is hyperbolic relative to a finite family of subgroups has finite decomposition complexity if and only if each subgroup has finite decomposition complexity.