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 ...
, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in 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 iden ...
acting
Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a broad r ...
on a set freely
generates a
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
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 that group.
History
The ping-pong argument goes back to the late 19th century and is commonly attributed
to
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
who used it to study subgroups of
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s, that is, of discrete groups of
isometries
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 ...
of the
hyperbolic 3-space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...
or, equivalently
Möbius transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form
f(z) = \frac
of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' ...
s of the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
. The ping-pong lemma was a key tool used by
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Life and ...
in his 1972 paper
[J. Tits]
''Free subgroups in linear groups.''
Journal of Algebra
''Journal of Algebra'' (ISSN 0021-8693) is an international mathematical research journal in algebra. An imprint of Academic Press, it is published by Elsevier. ''Journal of Algebra'' was founded by Graham Higman, who was its editor from 1964 to 1 ...
, vol. 20 (1972), pp. 250–270 containing the
proof of a famous result now known as the
Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Statement
The theorem, proven by Tits, is stated as follows.
Consequences
A linear group is not am ...
. The result states that a
finitely generated linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithf ...
is either
virtually
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 b ...
solvable or contains a free subgroup of
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
two. The ping-pong lemma and its variations are widely used in
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
and
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 ...
.
Modern versions of the ping-pong lemma can be found in many books such as Lyndon & Schupp,
[ Roger C. Lyndon and Paul E. Schupp. ''Combinatorial Group Theory.'' Springer-Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. ; Ch II, Section 12, pp. 167–169] de la Harpe,
[Pierre de la Harpe]
''Topics in geometric group theory.''
Chicago Lectures in Mathematics. University of Chicago Press, Chicago. ; Ch. II.B "The table-Tennis Lemma (Klein's criterion) and examples of free products"; pp. 25–41. Bridson & Haefliger
[Martin R. Bridson, and André Haefliger]
''Metric spaces of non-positive curvature.''
Grundlehren der Mathematischen Wissenschaften undamental Principles of Mathematical Sciences 319. Springer-Verlag, Berlin, 1999. ; Ch.III.Γ, pp. 467–468 and others.
Formal statements
Ping-pong lemma for several subgroups
This version of the ping-pong lemma ensures that several subgroups of a group
acting
Acting is an activity in which a story is told by means of its enactment by an actor or actress who adopts a character—in theatre, television, film, radio, or any other medium that makes use of the mimetic mode.
Acting involves a broad r ...
on a set generate a
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
. The following statement appears in Olijnyk and Suchchansky (2004), and the proof is from de la Harpe (2000).
Let ''G'' be a group acting on a set ''X'' and let ''H''
1, ''H''
2, ..., ''H''
''k'' be subgroups of ''G'' where ''k'' ≥ 2, such that at least one of these subgroups has
order greater than 2.
Suppose there exist
pairwise disjoint
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A ...
nonempty
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
subsets of such that the following holds:
*For any and for any in , we have .
Then
Proof
By the definition of free product, it suffices to check that a given (nonempty) reduced word represents a nontrivial element of
. Let
be such a word of length
, and let
where
for some
. Since
is reduced, we have
for any
and each
is distinct from the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of
. We then let
act on an element of one of the sets
. As we assume that at least one subgroup
has order at least 3,
without loss of generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate ...
we may assume that
has order at least 3. We first make the assumption that
and
are both 1 (which implies
). From here we consider
acting on
. We get the following chain of containments:
By the assumption that different
's are disjoint, we conclude that
acts nontrivially on some element of
, thus
represents a nontrivial element of
.
To finish the proof we must consider the three cases:
*if
, then let
(such an
exists since by assumption
has order at least 3);
*if
, then let
;
*and if
, then let
.
In each case,
after reduction becomes a reduced word with its first and last letter in
. Finally,
represents a nontrivial element of
, and so does
. This proves the claim.
The Ping-pong lemma for cyclic subgroups
Let ''G'' be a group acting on a set ''X''. Let ''a''
1, ...,''a''
''k'' be elements of ''G'' of infinite
order, where ''k'' ≥ 2. Suppose there exist disjoint nonempty subsets
of with the following properties:
* for ;
* for .
Then the subgroup
generated by ''a''
1, ..., ''a''
''k'' is
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
with free basis .
Proof
This statement follows as a
corollary
In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
of the version for general subgroups if we let and let .
Examples
Special linear group example
One can use the ping-pong lemma to prove
that the subgroup , generated by the
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and
is free of
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* H ...
two.
Proof
Indeed, let and be
cyclic subgroup
In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
In oth ...
s of generated by and accordingly. It is not hard to check that and are elements of infinite order in and that
and
Consider the standard
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of on by
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s. Put
and
It is not hard to check, using the above explicit descriptions of ''H''
1 and ''H''
2, that for every nontrivial we have and that for every nontrivial we have . Using the alternative form of the ping-pong lemma, for two subgroups, given above, we conclude that . Since the groups and are infinite
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in soc ...
, it follows that ''H'' is a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
of rank two.
Word-hyperbolic group example
Let be a
word-hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
which is
torsion-free, that is, with no nonidentity elements of finite
order. Let be two non-commuting elements, that is such that . Then there exists ''M'' ≥ 1 such that for any
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s , the subgroup is free of rank two.
Sketch of the proof[M. Gromov. ''Hyperbolic groups.'' Essays in group theory, pp. 75–263, Mathematical Sciences Research Institute Publications, 8, Springer, New York, 1987; ; Ch. 8.2, pp. 211–219.]
The group ''G'' acts on its ''hyperbolic boundary'' ∂''G'' by
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
s. It is known that if ''a'' in ''G'' is a nonidentity element then ''a'' has exactly two distinct fixed points, and in and that is an
attracting fixed point while is a
repelling fixed point.
Since and do not commute, basic facts about word-hyperbolic groups imply that , , and are four distinct points in . Take disjoint
neighborhoods
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...
, , , and of , , and in respectively.
Then the attracting/repelling properties of the fixed points of ''g'' and ''h'' imply that there exists such that for any integers , we have:
*
*
*
*
The ping-pong lemma now implies that is free of rank two.
Applications of the ping-pong lemma
*The ping-pong lemma is used in
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
s to study their so-called
Schottky subgroups. In the Kleinian groups context the ping-pong lemma can be used to show that a particular group of isometries of the
hyperbolic 3-space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...
is not just
free
Free may refer to:
Concept
* Freedom, having the ability to do something, without having to obey anyone/anything
* Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism
* Emancipate, to procur ...
but also
properly discontinuous
In mathematics, a group action on a space (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical ...
and
geometrically finite In geometry, a group of isometries of hyperbolic space is called geometrically finite if it has a well-behaved fundamental domain. A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite group ...
.
*Similar Schottky-type arguments are widely used 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 ...
, particularly for subgroups of
word-hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
s
and for
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
s of trees.
*The ping-pong lemma is also used for studying Schottky-type subgroups of
mapping class group
In mathematics, in the subfield of geometric topology, the mapping class group is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.
Mot ...
s of
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
s, where the set on which the mapping class group acts is the Thurston boundary of the
Teichmüller space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
. A similar argument is also utilized in the study of subgroups of the
outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...
of a free group.
*One of the most famous applications of the ping-pong lemma is in the proof of
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Life and ...
of the so-called
Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Statement
The theorem, proven by Tits, is stated as follows.
Consequences
A linear group is not am ...
for
linear group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a faithf ...
s.
(see also for an overview of Tits' proof and an explanation of the ideas involved, including the use of the ping-pong lemma).
*There are generalizations of the ping-pong lemma that produce not just
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
s but also
amalgamated free products and
HNN extension In mathematics, the HNN extension is an important construction of combinatorial group theory.
Introduced in a 1949 paper ''Embedding Theorems for Groups'' by Graham Higman, Bernhard Neumann, and Hanna Neumann, it embeds a given group ''G'' into an ...
s.
These generalizations are used, in particular, in the proof of
Maskit's Combination Theorem for Kleinian groups.
*There are also versions of the ping-pong lemma which guarantee that several elements in a group generate a
free semigroup In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero eleme ...
. Such versions are available both in the general context of a
group action
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 ...
on a set,
[Pierre de la Harpe]
''Topics in geometric group theory.''
Chicago Lectures in Mathematics. University of Chicago Press, Chicago. ; Ch. II.B "The table-Tennis Lemma (Klein's criterion) and examples of free products"; pp. 187–188. and for specific types of actions, e.g. in the context of linear groups, groups
acting on trees and others.
[Yves de Cornulier and Romain Tessera]
Quasi-isometrically embedded free sub-semigroups.
''Geometry & Topology
''Geometry & Topology'' is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sci ...
'', vol. 12 (2008), pp. 461–473; Lemma 2.1
References
{{reflist
See also
*
Free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
*
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
*
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
*
Tits alternative In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
Statement
The theorem, proven by Tits, is stated as follows.
Consequences
A linear group is not am ...
*
Word-hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
*
Schottky group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by .
Definition
Fix some point ''p'' on the Riemann sphere. Each Jordan curve not passing through ''p'' divides the Riemann sphere into two pieces, and we call ...
Lemmas in group theory
Discrete groups
Lie groups
Combinatorics on words