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 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 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, de la Harpe, Bridson & Haefliger 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), and the proof is from de la Harpe (2000).

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 $$\langle H_{1},\dots ,H_{k}\rangle =H_{1}\ast \dots \ast H_{k}.$$

Proof

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

By the assumption that different $X_{i}$'s are disjoint, we conclude that $w$ acts nontrivially on some element of $X_{2}$, thus $w$ represents a nontrivial element of $G$.

To finish the proof we must consider the three cases:

• if $\alpha _{1}=1,\,\alpha _{m}\neq 1$, then let $h\in H_{1}\setminus \{w_{1}^{-1},1\}$ (such an $h$ exists since by assumption $H_{1}$ has order at least 3);
• if $\alpha _{1}\neq 1,\,\alpha _{m}=1$, then let $h\in H_{1}\setminus \{w_{m},1\}$;
• and if $\alpha _{1}\neq 1,\,\alpha _{m}\neq 1$, then let $h\in H_{1}\setminus \{1\}$.

In each case, $hwh^{-1}$ after reduction becomes a reduced word with its first and last letter in $H_{1}$. Finally, $hwh^{-1}$ represents a nontrivial element of $G$, and so does $w$. 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

X₁⁺, ..., X_k⁺ and X₁^–, ..., X_k^–

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 that the subgroup H = ⟨A,B⟩ ≤ SL₂(Z), generated by the matrices $$A={\begin{pmatrix}1&2\\0&1\end{pmatrix}}$$ and $$B={\begin{pmatrix}1&0\\2&1\end{pmatrix}}$$ 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 $$H_{1}=\{A^{n}\mid n\in \mathbb {Z} \}=\left\{{\begin{pmatrix}1&2n\\0&1\end{pmatrix}}:n\in \mathbb {Z} \right\}$$ and $$H_{2}=\{B^{n}\mid n\in \mathbb {Z} \}=\left\{{\begin{pmatrix}1&0\\2n&1\end{pmatrix}}:n\in \mathbb {Z} \right\}.$$

Consider the standard action of SL₂(Z) on R² by linear transformations. Put $$X_{1}=\left\{{\begin{pmatrix}x\\y\end{pmatrix}}\in \mathbb {R} ^{2}:|x|>|y|\right\}$$ and $$X_{2}=\left\{{\begin{pmatrix}x\\y\end{pmatrix}}\in \mathbb {R} ^{2}:|x|<|y|\right\}.$$

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

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 and for automorphism groups of trees.
• 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. A similar argument is also utilized in the study of subgroups of the outer automorphism group of a free group.
• 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 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.
• 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, and for specific types of actions, e.g. in the context of linear groups, groups acting on trees and others.

See also

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