Quasi-isometry

In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.
The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov.
Definition.
Suppose that formula_1 is a (not necessarily continuous) function from one metric space formula_2 to a second metric space formula_3. Then formula_1 is called a "quasi-isometry" from formula_2 to formula_3 if there exist constants formula_7, formula_8, and formula_9 such that the following two properties both hold:
The two metric spaces formula_2 and formula_3 are called quasi-isometric if there exists a quasi-isometry formula_1 from formula_2 to formula_3.
A map is called a quasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarsely Lipschitz but may fail to be coarsely surjective). In other words, if through the map, formula_2 is quasi-isometric to a subspace of formula_3.
Two metric spaces "M1" and "M2" are said to be quasi-isometric, denoted formula_26, if there exists a quasi-isometry formula_27.
Examples.
The map between the Euclidean plane and the plane with the Manhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most formula_28. Note that there can be no isometry, since, for example, the points formula_29 are of equal distance to each other in Manhattan distance, but in the Euclidea plane, there are no 4 points that are of equal distance to each other.
The map formula_30 (both with the Euclidean metric) that sends every formula_31-tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance formula_32 of an integer tuple. In the other direction, the discontinuous function that rounds every tuple of real numbers to the nearest integer tuple is also a quasi-isometry: each point is taken by this map to a point within distance formula_32 of it, so rounding changes the distance between pairs of points by adding or subtracting at most formula_34.
Every pair of finite or bounded metric spaces is quasi-isometric. In this case, every function from one space to the other is a quasi-isometry.
Equivalence relation.
If formula_35 is a quasi-isometry, then there exists a quasi-isometry formula_36. Indeed, formula_37 may be defined by letting formula_11 be any point in the image of formula_1 that is within distance formula_17 of formula_10, and letting formula_37 be any point in formula_43.
Since the identity map is a quasi-isometry, and the composition of two quasi-isometries is a quasi-isometry, it follows that the property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.
Use in geometric group theory.
Given a finite generating set "S" of a finitely generated group "G", we can form the corresponding Cayley graph of "S" and "G". This graph becomes a metric space if we declare the length of each edge to be 1. Taking a different finite generating set "T" results in a different graph and a different metric space, however the two spaces are quasi-isometric. This quasi-isometry class is thus an invariant of the group "G". Any property of metric spaces that only depends on a space's quasi-isometry class immediately yields another invariant of groups, opening the field of group theory to geometric methods.
More generally, the Švarc–Milnor lemma states that if a group "G" acts properly discontinuously with compact quotient on a proper geodesic space "X" then "G" is quasi-isometric to "X" (meaning that any Cayley graph for "G" is). This gives new examples of groups quasi-isometric to each other: 
Quasigeodesics and the Morse lemma.
A "quasi-geodesic" in a metric space formula_44 is a quasi-isometric embedding of formula_45 into formula_46. More precisely a map formula_47 such that there exists formula_48 so that 
is called a formula_50-quasi-geodesic. Obviously geodesics (parametrised by arclength) are quasi-geodesics. The fact that in some spaces the converse is coarsely true, i.e. that every quasi-geodesic stays within bounded distance of a true geodesic, is called the "Morse Lemma" (not to be confused with the perhaps more widely known Morse lemma in differential topology). Formally the statement is:
It is an important tool in geometric group theory. An immediate application is that any quasi-isometry between proper hyperbolic spaces induces a homeomorphism between their boundaries. This result is the first step in the proof of the Mostow rigidity theorem.
Examples of quasi-isometry invariants of groups.
The following are some examples of properties of group Cayley graphs that are invariant under quasi-isometry:
Hyperbolicity.
A group is called "hyperbolic" if one of its Cayley graphs is a δ-hyperbolic space for some δ. When translating between different definitions of hyperbolicity, the particular value of δ may change, but the resulting notions of a hyperbolic group turn out to be equivalent.
Hyperbolic groups have a solvable word problem. They are biautomatic and automatic.: indeed, they are strongly geodesically automatic, that is, there is an automatic structure on the group, where the language accepted by the word acceptor is the set of all geodesic words.
Growth.
The growth rate of a group with respect to a symmetric generating set describes the size of balls in the group. Every element in the group can be written as a product of generators, and the growth rate counts the number of elements that can be written as a product of length "n".
According to Gromov's theorem, a group of polynomial growth is virtually nilpotent, i.e. it has a nilpotent subgroup of finite index. In particular, the order of polynomial growth formula_59 has to be a natural number and in fact formula_60.
If formula_61 grows more slowly than any exponential function, "G" has a subexponential growth rate. Any such group is amenable.
Ends.
The ends of a topological space are, roughly speaking, the connected components of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification.
The ends of a finitely generated group are defined to be the ends of the corresponding Cayley graph; this definition is independent of the choice of a finite generating set. Every finitely-generated infinite group has either 0,1, 2, or infinitely many ends, and Stallings theorem about ends of groups provides a decomposition for groups with more than one end.
If two connected locally finite graphs are quasi-isometric then they have the same number of ends. In particular, two quasi-isometric finitely generated groups have the same number of ends.
Amenability.
An amenable group is a locally compact topological group "G" carrying a kind of averaging operation on bounded functions that is invariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets of "G", was introduced by John von Neumann in 1929 under the German name "messbar" ("measurable" in English) in response to the Banach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.
In discrete group theory, where "G" has the discrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what proportion of "G" any given subset takes up.
If a group has a Følner sequence then it is automatically amenable.
Asymptotic cone.
An ultralimit is a geometric construction that assigns to a sequence of metric spaces "Xn" a limiting metric space. An important class of ultralimits are the so-called "asymptotic cones" of metric spaces. Let ("X","d") be a metric space, let "ω" be a non-principal ultrafilter on formula_62 and let "pn" ∈ "X" be a sequence of base-points. Then the "ω"–ultralimit of the sequence formula_63 is called the asymptotic cone of "X" with respect to "ω" and formula_64 and is denoted formula_65. One often takes the base-point sequence to be constant, "pn" = "p" for some "p ∈ X"; in this case the asymptotic cone does not depend on the choice of "p ∈ X" and is denoted by formula_66 or just formula_67.
The notion of an asymptotic cone plays an important role in geometric group theory since asymptotic cones (or, more precisely, their topological types and bi-Lipschitz types) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular. Asymptotic cones also turn out to be a useful tool in the study of relatively hyperbolic groups and their generalizations.