Gromov Hyperbolic Groups Admit Relatively Coarse Embedding Into Hilbert Spaces

In this paper,we mainly study some problems of coarse embedding in coarse geometry.With the rapid development of noncommutative geometry,the research on operator algebra is increasing,the index theory in coarse geometry is an important part of it.The index theory of coarse geometry is mainly used to solve the coarse Baum-Connes conjecture and the coarse Novikov conjecture.In 2000,G.Yu proved that if a discrete metric space with bounded geometry admits a coarse embedding into the Hilbert space,then the coarse Baum-Connes conjecture holds for it,so the coarse Novikov conjecture also holds for it.In 2006,Kasparov and G.Yu proved that if a discrete metric space with bounded geometry admits a coarse embedding into the uniformly convex Banach space,then the coarse Novikov conjecture holds for it.G.Yu proposed the definition of property A and proved that a discrete metric space with property A admits a coarse embedding into the Hilbert space.First of all,inspired by the above research results of coarse embedding theory,and combined with the concept of relatively hyperbolic groups,we give the definition of relatively coarse embedding for the first time:When G is a finitely generated group and H is a finitely generated subgroup of G,if there is a mapping f:G/H?H^?(g H-??_{g H}),Where H^?is a Hilbert space,and f is a coarse embedding,then f is a relatively coarse embedding.Secondly,according to the definition of bounded valency and bounded growth at some scale,we prove that the coned-off Cayley graph(?)of G with respect to H has bounded growth at some scale.Finally,it is proved that when G is a finitely generated group,H is a finite subgroup of G,and G is hyperbolic relative to H,G/H admits a relatively coarse embedding into the Hilbert space.The results of this paper are of great significance to the relative Novikov conjecture.