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^[1] 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 paper^[2] 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,^[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, ..., 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 nonempty subsets X₁, X₂, ...,X_k of X such that the following holds:

• For any i ≠ s and for any h in H_i, h ≠ 1 we have h(X_s) ⊆ X_i.

Then [math]\displaystyle{ \langle H_1,\dots, H_k\rangle=H_1\ast\dots \ast H_k. }[/math]

Proof

By the definition of free product, it suffices to check that a given (nonempty) reduced word represents a nontrivial element of [math]\displaystyle{ G }[/math]. Let [math]\displaystyle{ w }[/math] be such a word of length [math]\displaystyle{ m\geq 2 }[/math], and let [math]\displaystyle{ w = \prod_{i=1}^m w_i, }[/math] where [math]\displaystyle{ w_i \in H_{\alpha_i} }[/math] for some [math]\displaystyle{ \alpha_i \in \{1,\dots,k\} }[/math]. Since [math]\displaystyle{ w }[/math] is reduced, we have [math]\displaystyle{ \alpha_i \neq \alpha_{i+1} }[/math] for any [math]\displaystyle{ i = 1, \dots, m-1 }[/math] and each [math]\displaystyle{ w_i }[/math] is distinct from the identity element of [math]\displaystyle{ H_{\alpha_i} }[/math]. We then let [math]\displaystyle{ w }[/math] act on an element of one of the sets [math]\displaystyle{ X_i }[/math]. As we assume that at least one subgroup [math]\displaystyle{ H_i }[/math] has order at least 3, without loss of generality we may assume that [math]\displaystyle{ H_1 }[/math] has order at least 3. We first make the assumption that [math]\displaystyle{ \alpha_1 }[/math]and [math]\displaystyle{ \alpha_m }[/math] are both 1 (which implies [math]\displaystyle{ m \geq 3 }[/math]). From here we consider [math]\displaystyle{ w }[/math] acting on [math]\displaystyle{ X_2 }[/math]. We get the following chain of containments: [math]\displaystyle{ w(X_2) \subseteq \prod_{i=1}^{m-1} w_i(X_1) \subseteq \prod_{i=1}^{m-2} w_i(X_{\alpha_{m-1}}) \subseteq \dots \subseteq w_1(X_{\alpha_2}) \subseteq X_1. }[/math]

By the assumption that different [math]\displaystyle{ X_i }[/math]'s are disjoint, we conclude that [math]\displaystyle{ w }[/math] acts nontrivially on some element of [math]\displaystyle{ X_2 }[/math], thus [math]\displaystyle{ w }[/math] represents a nontrivial element of [math]\displaystyle{ G }[/math].

To finish the proof we must consider the three cases:

• if [math]\displaystyle{ \alpha_1 = 1,\,\alpha_m \neq 1 }[/math], then let [math]\displaystyle{ h\in H_1\setminus \{w_1^{-1},1\} }[/math] (such an [math]\displaystyle{ h }[/math] exists since by assumption [math]\displaystyle{ H_1 }[/math] has order at least 3);
• if [math]\displaystyle{ \alpha_1 \neq 1,\,\alpha_m=1 }[/math], then let [math]\displaystyle{ h\in H_1\setminus \{w_m,1\} }[/math];
• and if [math]\displaystyle{ \alpha_1\neq 1,\,\alpha_m\neq 1 }[/math], then let [math]\displaystyle{ h\in H_1\setminus \{1\} }[/math].

In each case, [math]\displaystyle{ hwh^{-1} }[/math] after reduction becomes a reduced word with its first and last letter in [math]\displaystyle{ H_1 }[/math]. Finally, [math]\displaystyle{ hwh^{-1} }[/math] represents a nontrivial element of [math]\displaystyle{ G }[/math], and so does [math]\displaystyle{ w }[/math]. This proves the claim.

The Ping-pong lemma for cyclic subgroups

Let G be a group acting on a set X. Let a₁, ...,a_k be elements of G of infinite order, where k ≥ 2. Suppose there exist disjoint nonempty subsets

of X with the following properties:

• a_i(X − X_i^–) ⊆ X_i⁺ for i = 1, ..., k;
• a_i⁻¹(X − X_i⁺) ⊆ X_i^– for i = 1, ..., k.

Then the subgroup H = ⟨a₁, ..., a_k⟩ ≤ G generated by a₁, ..., a_k is free with free basis {a₁, ..., a_k}.

Proof

This statement follows as a corollary of the version for general subgroups if we let X_i = X_i⁺ ∪ X_i⁻ and let H_i = ⟨a_i⟩.

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 [math]\displaystyle{ A = \begin{pmatrix}1 & 2\\ 0 &1 \end{pmatrix} }[/math] and [math]\displaystyle{ B = \begin{pmatrix}1 & 0\\ 2 &1 \end{pmatrix} }[/math] is free of rank two.

Proof

Indeed, let H₁ = ⟨A⟩ and H₂ = ⟨B⟩ be cyclic subgroups of SL₂(Z) generated by A and B accordingly. It is not hard to check that A and B are elements of infinite order in SL₂(Z) and that [math]\displaystyle{ H_1 = \{A^n \mid n\in \Z\} = \left\{\begin{pmatrix}1 & 2n\\ 0 & 1 \end{pmatrix} : n\in\Z\right\} }[/math] and [math]\displaystyle{ H_2 = \{B^n \mid n\in \Z\} = \left\{\begin{pmatrix}1 & 0\\ 2n & 1 \end{pmatrix} : n\in\Z\right\}. }[/math]

Consider the standard action of SL₂(Z) on R² by linear transformations. Put [math]\displaystyle{ X_1 = \left\{ \begin{pmatrix}x \\ y \end{pmatrix}\in \R^2 : |x|\gt |y|\right\} }[/math] and [math]\displaystyle{ X_2 = \left\{ \begin{pmatrix}x \\ y \end{pmatrix}\in \mathbb R^2 : |x|\lt |y|\right\}. }[/math]

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 word-hyperbolic group which is torsion-free, that is, with no nonidentity elements of finite order. Let g, h ∈ G be two non-commuting elements, that is such that gh ≠ hg. Then there exists M ≥ 1 such that for any integers n ≥ M, m ≥ M the subgroup H = ⟨gⁿ, h^m⟩ ≤ G is free of rank two.

Sketch of the proof^[6]

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, a^∞ and a^−∞ in ∂G and that a^∞ is an attracting fixed point while a^−∞ is a repelling fixed point.

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_–
• h^m(∂G – V_–) ⊆ V₊
• h^−m(∂G – V₊) ⊆ V_–

The ping-pong lemma now implies that H = ⟨gⁿ, h^m⟩ ≤ G 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^[6] and for automorphism groups of trees.^[7]
• 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 free products but also 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

See also

• Free group
• Free product
• Kleinian group
• Tits alternative
• Word-hyperbolic group
• Schottky group

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