Rough isometries and the parabolicity of riemannian manifolds

When we study non-compact complete riemannian manifolds, we observe that quasi-isometric deformations of their metrics do not alter global or qualita- tive properties of the manifolds: For example, two complete riemannian mani- folds quasi-isometric to each other obviously have the same volume growth rate.On the other hand, for a non-compact complete riemannian manifold, "attaching finitely many handles" (see Fig. 1) also preserves such geometric invariants of the manifold; in other words, we may say that a local topological deformation of the manifold does not exert essential influences on global geometry.Suggested by these observations, we introduced the notion of rough isometry in [10].