Ping-pong lemma
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
Modern versions of the ping-pong lemma can be found in many books such as Lyndon & Schupp,[3] de la Harpe,[1] Bridson & Haefliger[4] 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),[5] and the proof is from de la Harpe (2000).[1]
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 X1, X2, ...,Xk of X such that the following holds:
- For any i ≠ s and for any h in Hi, h ≠ 1 we have h(Xs) ⊆ Xi.
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
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 a1, ...,ak be elements of G of infinite order, where k ≥ 2. Suppose there exist disjoint nonempty subsets
of X with the following properties:
- ai(X − Xi–) ⊆ Xi+ for i = 1, ..., k;
- ai−1(X − Xi+) ⊆ Xi– for i = 1, ..., k.
Then the subgroup H = ⟨a1, ..., ak⟩ ≤ G generated by a1, ..., ak is free with free basis {a1, ..., ak}.
Proof
This statement follows as a corollary of the version for general subgroups if we let Xi = Xi+ ∪ Xi− and let Hi = ⟨ai⟩.
Examples
Special linear group example
One can use the ping-pong lemma to prove[1] that the subgroup H = ⟨A,B⟩ ≤ SL2(Z), generated by the matrices
Proof
Indeed, let H1 = ⟨A⟩ and H2 = ⟨B⟩ be
Consider the standard
It is not hard to check, using the above explicit descriptions of H1 and H2, that for every nontrivial g ∈ H1 we have g(X2) ⊆ X1 and that for every nontrivial g ∈ H2 we have g(X1) ⊆ X2. Using the alternative form of the ping-pong lemma, for two subgroups, given above, we conclude that H = H1 ∗ H2. Since the groups H1 and H2 are infinite cyclic, it follows that H is a free group of rank two.
Word-hyperbolic group example
Let G be a
Sketch of the proof[6]
The group G acts on its hyperbolic boundary ∂G by
Since g and h do not commute, basic facts about word-hyperbolic groups imply that g∞, g−∞, h∞ and h−∞ are four distinct points in ∂G. Take disjoint neighborhoods U+, U–, V+, and V– of g∞, g−∞, h∞ and h−∞ in ∂G respectively. Then the attracting/repelling properties of the fixed points of g and h imply that there exists M ≥ 1 such that for any integers n ≥ M, m ≥ M we have:
- gn(∂G – U–) ⊆ U+
- g−n(∂G – U+) ⊆ U–
- hm(∂G – V–) ⊆ V+
- h−m(∂G – V+) ⊆ V–
The ping-pong lemma now implies that H = ⟨gn, hm⟩ ≤ G is free of rank two.
Applications of the ping-pong lemma
- The ping-pong lemma is used in geometrically finite.
- Similar Schottky-type arguments are widely used in
- 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.[8] A similar argument is also utilized in the study of subgroups of the outer automorphism group of a free group.[9]
- 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.[2] (see also [10] 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 amalgamated free products and HNN extensions.[3] These generalizations are used, in particular, in the proof of Maskit's Combination Theorem for Kleinian groups.[11]
- 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,[12] and for specific types of actions, e.g. in the context of linear groups,[13] groups acting on trees[14] and others.[15]
References
- ^ ISBN 0-226-31719-6; Ch. II.B "The table-Tennis Lemma (Klein's criterion) and examples of free products"; pp. 25–41.
- ^ a b J. Tits. Free subgroups in linear groups. Journal of Algebra, vol. 20 (1972), pp. 250–270
- ^ ISBN 978-3-540-41158-1; Ch II, Section 12, pp. 167–169
- ISBN 3-540-64324-9; Ch.III.Γ, pp. 467–468
- ^ Andrij Olijnyk and Vitaly Suchchansky. Representations of free products by infinite unitriangular matrices over finite fields. International Journal of Algebra and Computation. Vol. 14 (2004), no. 5–6, pp. 741–749; Lemma 2.1
- ^ ISBN 0-387-96618-8; Ch. 8.2, pp. 211–219.
- ^ Alexander Lubotzky. Lattices in rank one Lie groups over local fields. Geometric and Functional Analysis, vol. 1 (1991), no. 4, pp. 406–431
- ^ 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, ISBN 978-0-8218-4227-0; 0-8218-4227-7
- ^ 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.
- ^ Pierre de la Harpe. Free groups in linear groups. L'Enseignement Mathématique (2), vol. 29 (1983), no. 1-2, pp. 129–144
- ^ Bernard Maskit.
Kleinian groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 287. Springer-Verlag, Berlin, 1988. ISBN 3-540-17746-9; Ch. VII.C and Ch. VII.E pp.149–156 and pp. 160–167
- ISBN 0-226-31719-6; Ch. II.B "The table-Tennis Lemma (Klein's criterion) and examples of free products"; pp. 187–188.
- ^ 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
- ISBN 978-0-8218-3158-8; page 2, Lemma 3.1
- ^ Yves de Cornulier and Romain Tessera. Quasi-isometrically embedded free sub-semigroups. Geometry & Topology, vol. 12 (2008), pp. 461–473; Lemma 2.1
See also
- Free group
- Free product
- Kleinian group
- Tits alternative
- Word-hyperbolic group
- Schottky group