Research On Growth Types Of Some Large Scale Geometry Functions

As one of the central problems of index theory,coarse Baum-Connes conjecture gives a method to calculate the generalized Fredholm index of elliptical differential operators on noncompact smooth manifolds.Since Property A implies that coarse Baum-Connes conjecture holds,it is significant in index theory.Property A has many equivalent descriptions,especially two important ones:metric sparsification property and operator norm localization property,which depict Property A from the view of measure and operator norm,respectively.In this thesis,we define functions from[1,?)to^?10?relative to Propety A,metric sparsification property and operator norm localization property,respectively.We discuss how these functions be controlled under coarse equivalence and the coarse invariant of growth types.In the first place,for the most special case that coarse equivalence and quasi-isometry are equivalent,taking metric graph for instance,we prove that the growth type of the function relative to Property A is a coarse invariant.In the second place,for uniformly discrete metric spaces with bounded geometry in general,we show how the function relative to Property A be controlled under coarse equivalence.Moreover,for a special kind of spaces,the growth type of the function relative to Property A is a coarse invariant.Since Propety A,metric sparsification property and operator norm localization property are equivalent,above conclusions hold for metric sparsification property and operator norm localization property as well.At last,for a fixed Property A space,we discuss how these three functions control each other.Furthermore,for a special kind of spaces,the growth types of these functions are in the same.