Inmathematics, aquasi-isometry is afunction between twometric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces arequasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like anequivalence relation on theclass of metric spaces.

The concept of quasi-isometry is especially important ingeometric group theory, following the work ofGromov.^[1]

Suppose that${\displaystyle f}$  is a (not necessarily continuous) function from one metric space${\displaystyle (M_1,d_1)}$  to a second metric space${\displaystyle (M_2,d_2)}$ . Then${\displaystyle f}$  is called aquasi-isometry from${\displaystyle (M_1,d_1)}$  to${\displaystyle (M_2,d_2)}$  if there exist constants${\displaystyle A\ge 1}$ ,${\displaystyle B\ge 0}$ , and${\displaystyle C\ge 0}$  such that the following two properties both hold:^[2]

1. For every two points${\displaystyle x}$  and${\displaystyle y}$  in${\displaystyle M_1}$ , the distance between their images is up to the additive constant${\displaystyle B}$  within a factor of${\displaystyle A}$  of their original distance. More formally:

   ${\displaystyle \mathrm{\forall }x,y\in M_1:\frac{1}{A}{d}_1(x,y)-B\le d_2(f(x),f(y))\le A{d}_1(x,y)+B.}$ 

2. Every point of${\displaystyle M_2}$  is within the constant distance${\displaystyle C}$  of an image point. More formally:

   ${\displaystyle \mathrm{\forall }z\in M_2:\mathrm{\exists }x\in M_1:d_2(z,f(x))\le C.}$ 

The two metric spaces${\displaystyle (M_1,d_1)}$  and${\displaystyle (M_2,d_2)}$  are calledquasi-isometric if there exists a quasi-isometry${\displaystyle f}$  from${\displaystyle (M_1,d_1)}$  to${\displaystyle (M_2,d_2)}$ .

A map is called aquasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarselyLipschitz but may fail to be coarsely surjective). In other words, if through the map,${\displaystyle (M_1,d_1)}$  is quasi-isometric to a subspace of${\displaystyle (M_2,d_2)}$ .

Two metric spacesM₁ andM₂ are said to bequasi-isometric, denoted${\displaystyle M_1\underset{q.i.}{\sim }{M}_2}$ , if there exists a quasi-isometry${\displaystyle f:M_1\to M_2}$ .

The map between theEuclidean plane and the plane with theManhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most${\displaystyle \sqrt{2}}$ . Note that there can be no isometry, since, for example, the points${\displaystyle (1,0),(-1,0),(0,1),(0,-1)}$  are of equal distance to each other in Manhattan distance, but in the Euclidean plane, there are no 4 points that are of equal distance to each other.

The map${\displaystyle f:{\mathbb{Z}}^n\to {\mathbb{R}}^n}$  (both with theEuclidean metric) that sends every${\displaystyle n}$ -tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance${\displaystyle \sqrt{n/4}}$  of an integer tuple. In the other direction, the discontinuous function thatrounds 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${\displaystyle \sqrt{n/4}}$  of it, so rounding changes the distance between pairs of points by adding or subtracting at most${\displaystyle 2\sqrt{n/4}}$ .

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.

If${\displaystyle f:M_1\mapsto M_2}$  is a quasi-isometry, then there exists a quasi-isometry${\displaystyle g:M_2\mapsto M_1}$ . Indeed,${\displaystyle g(x)}$  may be defined by letting${\displaystyle y}$  be any point in the image of${\displaystyle f}$  that is within distance${\displaystyle C}$  of${\displaystyle x}$ , and letting${\displaystyle g(x)}$  be any point in${\displaystyle f^{-1}(y)}$ .

Since theidentity map is a quasi-isometry, and thecomposition of two quasi-isometries is a quasi-isometry, it follows that the property of being quasi-isometric behaves like anequivalence relation on the class of metric spaces.

Given a finitegenerating setS of a finitely generatedgroupG, we can form the correspondingCayley graph ofS andG. This graph becomes a metric space if we declare the length of each edge to be 1. Taking a different finite generating setT results in a different graph and a different metric space, however the two spaces are quasi-isometric.^[3] This quasi-isometry class is thus aninvariant of the groupG. 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, theŠvarc–Milnor lemma states that if a groupG actsproperly discontinuously with compact quotient on a proper geodesic spaceX thenG is quasi-isometric toX (meaning that any Cayley graph forG is). This gives new examples of groups quasi-isometric to each other:

• IfG' is a subgroup of finiteindex inG thenG' is quasi-isometric toG;
• IfG andH are the fundamental groups of two compacthyperbolic manifolds of the same dimensiond then they are both quasi-isometric to the hyperbolic spaceH^d and hence to each other; on the other hand there are infinitely many quasi-isometry classes of fundamental groups of finite-volume.^[4]

Aquasi-geodesic in a metric space${\displaystyle (X,d)}$  is a quasi-isometric embedding of${\displaystyle \mathbb{R}}$  into${\displaystyle X}$ . More precisely a map${\displaystyle \varphi :\mathbb{R}\to X}$  such that there exists${\displaystyle C,K>0}$  so that

${\displaystyle \mathrm{\forall }s,t\in \mathbb{R}:C^{-1}|s-t|-K\le d(\varphi (t),\varphi (s))\le C|s-t|+K}$ 

is called a${\displaystyle (C,K)}$ -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 theMorse Lemma (not to be confused with theMorse lemma in differential topology). Formally the statement is:

Let${\displaystyle \delta ,C,K>0}$  and${\displaystyle X}$  a properδ-hyperbolic space. There exists${\displaystyle M}$  such that for any${\displaystyle (C,K)}$ -quasi-geodesic${\displaystyle \varphi }$  there exists a geodesic${\displaystyle L}$  in${\displaystyle X}$  such that${\displaystyle d(\varphi (t),L)\le M}$  for all${\displaystyle t\in \mathbb{R}}$ .

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 theMostow rigidity theorem.

Furthermore, this result has found utility in analyzing user interaction design in applications similar toGoogle Maps.^[5]

The following are some examples of properties of group Cayley graphs that are invariant under quasi-isometry:^[2]

A group is calledhyperbolic 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 solvableword problem. They arebiautomatic andautomatic.:^[6] indeed, they arestrongly 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.

Thegrowth rate of agroup with respect to a symmetricgenerating 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 lengthn.

According toGromov's theorem, a group of polynomial growth isvirtually nilpotent, i.e. it has anilpotentsubgroup of finiteindex. In particular, the order of polynomial growth${\displaystyle k_0}$  has to be anatural number and in fact${\displaystyle \mathrm{\#}(n)\sim n^{k_0}}$ .

If${\displaystyle \mathrm{\#}(n)}$  grows more slowly than any exponential function,G has asubexponential growth rate. Any such group isamenable.

Theends of atopological space are, roughly speaking, theconnected components of the “ideal boundary” of the space. That is, each end represents a topologically distinct way to move toinfinity within the space. Adding a point at each end yields acompactification of the original space, known as theend compactification.

The ends of afinitely generated group are defined to be the ends of the correspondingCayley 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, andStallings 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.^[7] In particular, two quasi-isometric finitely generated groups have the same number of ends.

Anamenable group is alocally compacttopological groupG carrying a kind of averaging operation on bounded functions that isinvariant under translation by group elements. The original definition, in terms of a finitely additive invariant measure (or mean) on subsets ofG, was introduced byJohn von Neumann in 1929 under theGerman name "messbar" ("measurable" in English) in response to theBanach–Tarski paradox. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.^[8]

Indiscrete group theory, whereG has thediscrete topology, a simpler definition is used. In this setting, a group is amenable if one can say what proportion ofG any given subset takes up.

If a group has aFølner sequence then it is automatically amenable.

Anultralimit is a geometric construction that assigns to a sequence ofmetric spacesX_n a limiting metric space. An important class of ultralimits are the so-calledasymptotic cones of metric spaces. Let (X,d) be a metric space, letω be a non-principal ultrafilter on${\displaystyle \mathbb{N}}$  and letp_n ∈ X be a sequence of base-points. Then theω–ultralimit of the sequence${\displaystyle (X,\frac{d}{n},p_n)}$  is called the asymptotic cone ofX with respect toω and${\displaystyle (p_n{)}_n}$  and is denoted${\displaystyle Con{e}_{\omega }(X,d,(p_n{)}_n)}$ . One often takes the base-point sequence to be constant,p_n =p for somep ∈ X; in this case the asymptotic cone does not depend on the choice ofp ∈ X and is denoted by${\displaystyle Con{e}_{\omega }(X,d)}$  or just${\displaystyle Con{e}_{\omega }(X)}$ .

The notion of an asymptotic cone plays an important role ingeometric group theory since asymptotic cones (or, more precisely, theirtopological types andbi-Lipschitz types) provide quasi-isometry invariants of metric spaces in general and of finitely generated groups in particular.^[9] Asymptotic cones also turn out to be a useful tool in the study ofrelatively hyperbolic groups and their generalizations.^[10]

• Isometry
• Coarse structure