Quasi-isometries
Quasi-isometries [ edit ] Main article: Quasi-isometry A quasi-isometry is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example, � 2 and its subspace � 2 are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in geometric group theory : the Švarc–Milnor lemma states that all spaces on which a group acts geometrically are quasi-isometric. [15] Formally, the map � : � 1 → � 2 is a quasi-isometric embedding if there exist constants A ≥ 1 and B ≥ 0 such that 1 � � 2 ( � ( � ) , � ( � ) ) − � ≤ � 1 ( � , � ) ≤ � � 2 ( � ( � ) , � ( � ) ) + � for all � , � ∈ � 1 . It is a quasi-isometry if in addition it is quasi-surjective , i.e. there is a constant C ≥ 0 such that every point in � 2 is at distance at most C from some point in the image � ( � 1 ) . Notions of metric space equivalence [ edit ] See also