Ping-Pong Lemma

1. Why it is called "Ping-Pong Lemma"? 

 
   All the versions of "Ping-Pong Lemma" are constructed in the same manner: if you want to prove that a group is a free group, it is sufficient to prove that that group can "ping-pong" an element from a "region" to another "region" and ensure that it won't go back to the original "region". This is similar to the case when you "ping-pong" the ball when playing table tennis, where you need to move the ball from your side to the rival's side and prevent the ball from tracing back to your side. 

   This lemma is sometimes called the Schottky lemma (named after the German mathematician Friedrich Schottky) or Klein's criterion (named after the German mathematician Felix Klein).  

2. Different versions of "Ping-Pong Lemma"

   There have been various versions of "Ping-Pong Lemma" mentioned in different textbooks. As Johanna Mangahas wrote in "Office Hours with a Geometric Group Theorist", "This elegent lemma has several variants.". Here, we mention some of these versions. The following version is taken from "Groups, Graphs and Trees" written by John Meier. 

   
   This version can be proven as follows: Consider the free group generated by S and a non-identity element in this group. By 1., each letter in the "word" representing this element send p to one of the sets X(s) (s in S), and this element must send p to an element in the union of all sets of the form X(s). Therefore, this element can't be the identity element of G. Since this holds for all elements of the free group generated by S, we conclude that G is indeed a free group with basis S. 

   Meanwhile, the version in "Office Hours with a Geometric Group Theorist" written by Johanna Mangahas doesn't require the action to send an element outside the union of all sets of the form X(s) to an element inside. In other words, it only considers sending "balls" between different player's side instead of requiring the "balls" outside the playing area to go back. We might think of the element p as someone whose job is to pick the ball whenever it (accidentally) goes far away from the playing area. 

   
   The proof of this version requires the use of conjugation. In other words, starting from a word, we must prove that there is a conjugation of that word which has the desired form and enables us to "send the ball from one side to another side".  

    A (generalized) version of ping-pong lemma can also be used to prove that a group is a free product of a family of subgroups generating it. Before showing this version, the definition of a free product of groups will be given. Briefly speaking, the free product of a family of groups G(1), G(2), ..., G(n) is the group generated by G(1), G(2), ..., G(n) without adding any additional relations. For example, the familiar free group is obtained when G(1), G(2), ..., G(n) are all isomorphic to the additive group of integers.

   Here is the "free product" version, which appeared in a paper written by Jacques Tits (1930 - 2021, Abel Prize 2008), titled "Free Subgroups in Linear Groups" (1972). This version was similar to the first version appearing in John Meier's book in the sense that there must be a point p outside, thus the proof is also similar.