In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a quasi-isometry is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between two
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s 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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
on the
class
Class or The Class may refer to:
Common uses not otherwise categorized
* Class (biology), a taxonomic rank
* Class (knowledge representation), a collection of individuals or objects
* Class (philosophy), an analytical concept used differentl ...
of metric spaces.
The concept of quasi-isometry is especially important in
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
, following the work of
Gromov
Gromov (russian: Громов) is a Russian male surname, its feminine counterpart is Gromova (Громова).
Gromov may refer to:
* Alexander Georgiyevich Gromov (born 1947), Russian politician and KGB officer
* Alexander Gromov (born 1959), R ...
.
Definition
Suppose that
is a (not necessarily continuous) function from one metric space
to a second metric space
. Then
is called a ''quasi-isometry'' from
to
if there exist constants
,
, and
such that the following two properties both hold:
[P. de la Harpe, ''Topics in geometric group theory''. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ]
#For every two points
and
in
, the distance between their images is up to the additive constant
within a factor of
of their original distance. More formally:
#:
#Every point of
is within the constant distance
of an image point. More formally:
#:
The two metric spaces
and
are called quasi-isometric if there exists a quasi-isometry
from
to
.
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,
is quasi-isometric to a subspace of
.
Two metric spaces ''M
1'' and ''M
2'' are said to be quasi-isometric, denoted
, if there exists a quasi-isometry
.
Examples
The map between the
Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
and the plane with the
Manhattan distance
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences ...
that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most
. Note that there can be no isometry, since, for example, the points
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
(both with the
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
) that sends every
-tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance
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
of it, so rounding changes the distance between pairs of points by adding or subtracting at most
.
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
is a quasi-isometry, then there exists a quasi-isometry
. Indeed,
may be defined by letting
be any point in the image of
that is within distance
of
, and letting
be any point in
.
Since the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
is a quasi-isometry, and the
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of two quasi-isometries is a quasi-isometry, it follows that the property of being quasi-isometric behaves like an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
on the class of metric spaces.
Use in geometric group theory
Given a finite
generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
''S'' of a finitely generated
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'', we can form the corresponding
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
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
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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 In the mathematical subject of geometric group theory, the Švarc–Milnor lemma (sometimes also called Milnor–Švarc lemma, with both variants also sometimes spelling Švarc as Schwarz) is a statement which says that a group G, equipped with ...
states that if a group ''G'' acts
properly discontinuously
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
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:
* If ''G' '' is a subgroup of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
in ''G'' then ''G' '' is quasi-isometric to ''G'';
* If ''G'' and ''H'' are the fundamental groups of two compact
hyperbolic manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, res ...
s of the same dimension ''d'' then they are both quasi-isometric to the hyperbolic space H
''d'' and hence to each other; on the other hand there are infinitely many quasi-isometry classes of fundamental groups of finite-volume.
Quasigeodesics and the Morse lemma
A ''quasi-geodesic'' in a metric space
is a quasi-isometric embedding of
into
. More precisely a map
such that there exists
so that
:
is called a
-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 mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differenti ...
in differential topology). Formally the statement is:
:''Let
and
a proper
δ-hyperbolic space In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properti ...
. There exists
such that for any
-quasi-geodesic
there exists a geodesic
in
such that
for all
. ''
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
with respect to a symmetric
generating set
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
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
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group ''G'' is said to ...
, i.e. it has a
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the class ...
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of finite
index
Index (or its plural form indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on a Halo megastru ...
. In particular, the order of polynomial growth
has to be a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
and in fact
.
If
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
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
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
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
within the space. Adding a point at each end yields a
compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
of the original space, known as the end compactification.
The ends of a
finitely generated group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses of s ...
are defined to be the ends of the corresponding
Cayley graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cay ...
; 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 In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group ''G'' has more than one end if and only if the group ''G'' admits a nontrivial decomposition as an amalgamated free produ ...
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 In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...
''G'' carrying a kind of averaging operation on bounded functions that is
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
in 1929 under the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
name "messbar" ("measurable" in English) in response to the
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be p ...
. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun.
In
discrete group theory, where ''G'' has the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, 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 space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
s ''X
n'' 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
and let ''p
n'' ∈ ''X'' be a sequence of base-points. Then the ''ω''–ultralimit of the sequence
is called the asymptotic cone of ''X'' with respect to ''ω'' and
and is denoted
. One often takes the base-point sequence to be constant, ''p
n'' = ''p'' for some ''p ∈ X''; in this case the asymptotic cone does not depend on the choice of ''p ∈ X'' and is denoted by
or just
.
The notion of an asymptotic cone plays an important role in
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
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.
[John Roe. ''Lectures on Coarse Geometry.'' ]American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
, 2003. Asymptotic cones also turn out to be a useful tool in the study of
relatively hyperbolic group In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete ...
s and their generalizations.
[ Cornelia Druţu and Mark Sapir (with an Appendix by ]Denis Osin Denis Osin is a mathematician at Vanderbilt University working in geometric group theory and geometric topology.
Career
Osin received a PhD at Moscow State University in 1999 under the supervision of Aleksandr Olshansky. He worked at the Fina ...
and ), ''Tree-graded spaces and asymptotic cones of groups.'' Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, Volume 44 (2005), no. 5, pp. 959–1058.
See also
*
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
*
Coarse structure In the mathematical fields of geometry and topology, a coarse structure on a set ''X'' is a collection of subsets of the cartesian product ''X'' × ''X'' with certain properties which allow the ''large-scale structure'' of metric spaces and topolo ...
References
{{reflist
Geometric group theory
Metric geometry
Equivalence (mathematics)