Quasi-isometry

The notion of quasi- isometry is used in mathematics to investigate the " rough" global geometry of metric spaces. He plays in many areas of geometry, analysis, and geometric group theory an important role, for example in the theory of hyperbolic groups or in evidence of rigidity sets.

Definition

Let ( M1, d1) and (M2, d2 ) are two metric spaces. A (not necessarily constant ) f: M1 → M2 is a quasi- isometric, if there are constants A ≥ 1 and B ≥ 0, such that

And a constant C ≥ 0 such that for every u in M2 x in M1 are with

The spaces M1 and M2 are called quasi- isometric if there is a quasi- isometry f: M1 → M2 is.

Examples

Every bounded metric space is quasi- isometric to the point.

The embedding is a quasi- isometry for the Euclidean metric and. It is in the above definition, A = 1, B = 0 and C = 1.

The assigned to different finite generating systems S1, S2 of a group of Cayley graphs are quasi- isometric.

Shvartz - Milnor Lemma: If a finitely generated group G kokompakt and properly discontinuous acts by isometries on a Riemannian manifold Y, then ( the Cayley graph of ) G quasi- isometric to Y.

With one obtains in particular: The fundamental group of a compact Riemannian manifold is quasi- isometric to the universal cover.