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. 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. 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 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.