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, ..., 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 ⟨ H 1 , … , H k ⟩ = H 1 ∗ ⋯ ∗ H k . {\displaystyle \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 {\displaystyle G} . Let w {\displaystyle w} be such a word of length m ≥ 2 {\displaystyle m\geq 2} , and let w = ∏ i = 1 m w i , {\displaystyle w=\prod _{i=1}^{m}w_{i},} where w i ∈ H α i {\textstyle w_{i}\in H_{\alpha _{i}}} for some α i ∈ { 1 , … , k } {\textstyle \alpha _{i}\in \{1,\dots ,k\}} . Since w {\textstyle w} is reduced, we have α i ≠ α i + 1 {\displaystyle \alpha _{i}\neq \alpha _{i+1}} for any i = 1 , … , m − 1 {\displaystyle i=1,\dots ,m-1} and each w i {\displaystyle w_{i}} is distinct from the identity element of H α i {\displaystyle H_{\alpha _{i}}} . We then let w {\displaystyle w} act on an element of one of the sets X i {\textstyle X_{i}} . As we assume that at least one subgroup H i {\displaystyle H_{i}} has order at least 3, without loss of generality we may assume that H 1 {\displaystyle H_{1}} has order at least 3. We first make the assumption that α 1 {\displaystyle \alpha _{1}} and α m {\displaystyle \alpha _{m}} are both 1 (which implies m ≥ 3 {\displaystyle m\geq 3} ). From here we consider w {\displaystyle w} acting on X 2 {\displaystyle X_{2}} . We get the following chain of containments:
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