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[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.
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: Then
By the definition of free product, it suffices to check that a given (nonempty) reduced word represents a nontrivial element of
is distinct from the identity element of
has order at least 3, without loss of generality we may assume that
represents a nontrivial element of
after reduction becomes a reduced word with its first and last letter in
represents a nontrivial element of
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: Then the subgroup H = ⟨a1, ..., ak⟩ ≤ G generated by a1, ..., ak is free with free basis {a1, ..., ak}.
This statement follows as a corollary of the version for general subgroups if we let Xi = Xi+ ∪ Xi− and let Hi = ⟨ai⟩.
One can use the ping-pong lemma to prove[1] that the subgroup H = ⟨A,B⟩ ≤ SL2(Z), generated by the matrices
It is not hard to check that A and B are elements of infinite order in SL2(Z) and that
Consider the standard action of SL2(Z) on R2 by linear transformations.
Using the alternative form of the ping-pong lemma, for two subgroups, given above, we conclude that H = H1 ∗ H2.
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, hm⟩ ≤ G is free of rank two.
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.
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: The ping-pong lemma now implies that H = ⟨gn, hm⟩ ≤ G is free of rank two.