Fibred Cofinitely-coarse Embeddability Of Box Families

In this paper,we focused on some problems in large scale geometry.The index theory of coarse geometry is the important researching branch in the field of Noncommutative Geometry in recent 2 decades.Coarse Baum-Connes conjecture and coarse Novikov conjecture are central issues of index theory in large scale geometry.It is of great significance to study embedded spaces and embedding methods for coarse geometry Novikov conjecture.In 2000,G.Yu proved that the coarse Baum-Connes conjecture which implies the coarse Novikov conjecture holds for any discrete metric space with bounded geometry admitting a coarse embedding into Hilbert space^[134].In2012,in^[34],X.Chen,Q.Wang and G.Yu proved that the maximal coarse Baum-connes conjecture which implies the maximal coarse Novikov conjecture holds for bounded geometry spaces which admit a fibred coarse embedding into a Hilbert space.In the first,inspired by^{[4][33][73][98]},we showed that a countable,residually amenable group admits a proper isometric affine action on some uniformly convex Banach space if and only if one?or equivalently,all?of its box families admits a fibred cofinitely-coarse embedding into some uniformly convex Banach space.Moreover,the fibred coarse embeddability of a warped cone implies the Haagerup property of the appropriate group.A direct proof for the result is presented.Furthermore,we defined the coarse version of finite APC-decomposition complexity in the language of coarse spaces.Under this significance,we study some permanence properties of finite APC-decomposition complexity and have proved that finite coarse APC-decomposition complexity implies coarse property A.In the end,we showed that the strong embeddability has fibering permanence property and is preserved under the direct limit for the metric space.Additionally,we proved the hereditary of strong embeddability under the coarse quasi-action of a group.In particular,we showed that strong embeddability is equivalent to property A for metric spaces with bounded geometry.