In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group.
History
The ping-pong argument goes back to the late 19th century and is commonly attributed to Felix Klein who used it to study subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalently Möbius transformations of the Riemann sphere. The ping-pong lemma was a key tool used by Jacques Tits in his 1972 paperJ. Tits. Free subgroups in linear groups. Journal of Algebra, vol. 20 (1972), pp. 250–270 containing the proof of a famous result now known as the Tits alternative. The result states that a finitely generated linear group is either virtually solvable or contains a free subgroup of rank two. The ping-pong lemma and its variations are widely used in geometric topology and geometric group theory.
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 & HaefligerMartin R. Bridson, and André Haefliger. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental 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 on a set generate a free product. The following statement appears in Olijnyk and Suchchansky (2004),; see Lemma 2.1 and the proof is from de la Harpe (2000).
Let G be a group acting on a set X and let H1, H2, ..., Hk 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 nonempty 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 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 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 infinite cyclic subgroups
Let G be a group acting on a set X. Let a1, ...,ak 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 a1, ..., ak is free with free basis .
Proof This statement follows as a corollary 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 and is free of rank two.
Proof Indeed, let and be cyclic subgroups 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 of on by linear transformations. Put
and
It is not hard to check, using the above explicit descriptions of H1 and H2, 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, it follows that H is a free group of rank two.
Word-hyperbolic group example
Let be a word-hyperbolic group 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 integers , the subgroup is free of rank two.
Sketch of the proof Source: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 homeomorphisms. 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 , , , 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 groups 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 is not just free but also properly discontinuous and geometrically finite. Similar Schottky-type arguments are widely used in geometric group theory, particularly for subgroups of word-hyperbolic groups and for automorphism groups of trees.Alexander Lubotzky. Lattices in rank one Lie groups over local fields. Geometric and Functional Analysis, vol. 1 (1991), no. 4, pp. 406–431 The ping-pong lemma is also used for studying Schottky-type subgroups of mapping class groups of Riemann surfaces, where the set on which the mapping class group acts is the Thurston boundary of the Teichmüller space.Richard P. Kent, and Christopher J. Leininger. Subgroups of mapping class groups from the geometrical viewpoint. In the tradition of Ahlfors-Bers. IV, pp. 119–141, Contemporary Mathematics series, 432, American Mathematical Society, Providence, RI, 2007; ; 0-8218-4227-7 A similar argument is also utilized in the study of subgroups of the outer automorphism group of a free group.M. Bestvina, M. Feighn, and M. Handel. Laminations, trees, and irreducible automorphisms of free groups. Geometric and Functional Analysis, vol. 7 (1997), no. 2, pp. 215–244. One of the most famous applications of the ping-pong lemma is in the proof of Jacques Tits of the so-called Tits alternative for linear groups. (see also Pierre de la Harpe. Free groups in linear groups. L'Enseignement Mathématique (2), vol. 29 (1983), no. 1-2, pp. 129–144 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 products but also amalgamated free products and HNN extensions. These generalizations are used, in particular, in the proof of Maskit's Combination Theorem for Kleinian groups.Bernard Maskit. Kleinian groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 287. Springer-Verlag, Berlin, 1988. ; Ch. VII.C and Ch. VII.E pp.149–156 and pp. 160–167 There are also versions of the ping-pong lemma which guarantee that several elements in a group generate a free semigroup. Such versions are available both in the general context of a group action 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,Alex Eskin, Shahar Mozes and Hee Oh. On uniform exponential growth for linear groups. Inventiones Mathematicae. vol. 60 (2005), no. 1, pp.1432–1297; Lemma 2.2 groups acting on treesRoger C. Alperin and Guennadi A. Noskov. Uniform growth, actions on trees and GL2. Computational and Statistical Group Theory:AMS Special Session Geometric Group Theory, April 21–22, 2001, Las Vegas, Nevada, AMS Special Session Computational Group Theory, April 28–29, 2001, Hoboken, New Jersey. (Robert H. Gilman, Vladimir Shpilrain, Alexei G. Myasnikov, editors). American Mathematical Society, 2002. ; page 2, Lemma 3.1 and others.Yves de Cornulier and Romain Tessera. Quasi-isometrically embedded free sub-semigroups. Geometry & Topology, vol. 12 (2008), pp. 461–473; Lemma 2.1
References
See also Free group Free product Kleinian group Tits alternative Word-hyperbolic group Schottky group