Generalizations And Permanence Of Property A In Coarse Geometry

Property A and coarse embeddability are two important notions of coarse geometry and high index theory.In 1993,Gromov introduced a definition of coarse(uniform) embeddability,and hinted that coarse embedding into Hilbert spaces or Banach spaces may have an important significant in solving the coarse Novikov conjecture.In 2000,G. Yu introduced property A of a discrete metric space which implies a coarse embedding into Hilbert spaces,and consequently,the coarse Baum-Cones conjecture for metric spaces,and the Novikov conjecture if metric space is the finite generated group with the word length metric.Until now,we have had variety equivalent characterization of property A,the most important one is given by Higson-Roe,which characterized by support condition and convergence condition.Nowadays,however,several type of metric spaces do not have property A and do not admit a coarse embedding into Hilbert space,such as Expander graph.In 2006,Kasparov and G.Yu proved that if a bounded geometry metric space admits a coarse embedding merely into uniformly convex Banach space,then the coarse Novikov conjecture is true for this metric space.In this paper,in the above context,we introduced a generalized version of Yu's property A,called generalized property A,which associate to uniformly convex Banach spaces as motivated by support condition and convergence condition.We proved that a countable discrete metric space with generalized property A admits a coarse embedding into uniformly convex Banach space,and studied the coarse invariance of generalized property A and the permanence properties of generalized property A with respect to relatively hyperbolic group and finite graph of group.On the other hand,Guentner,Tessera and G.Yu introduced the notion of finite decompositions complexity as motivated by the properties of finite asymptotic dimension, which can be regarded as an operation of metric spaces.At the same time,they proved that the stable Borel conjecture holds for aspherical manifolds whose fundamental groups have finite decompositions complexity.In this paper,we regarded the finite decompositions complexity as an operation of metric spaces,and proved that property A and coarse embeddability are invariants under finite decompositions complexity. To do this,we introduced two new notions by replacing the requirement "the family of metric spaces {X_{i1j1…injn}}_{i1,j1.…,in,jn} is unformly bounded" in the definition of finite decompositions complexity by the requirements that this family has "equiproperty A" or "equi-coarse embeddibility" respectively.In the following,we proved a finer "quantitative version of finite union theorem" for property A and a finer "finite decompositions complexity" for coarse embeddabilty,and applied them to proved the permanence properties of property A and coarse embeddability of metric spaces under finite decompositions complexity.