TITLE: What is the intution behind the ping-pong lemma?
QUESTION [17 upvotes]: Is the ping-pong lemma a difficult characterization of free groups? Or is it just me? Does someone have a nice intuition about its idea or should I carry on staring at the statement?

REPLY [3 votes]: The basic idea is very simple and nice. Suppose that $A$ and $B$ are matrices, or more generally maps, such that $Av$ and $Bw$ are different for all $v$ and $w$. The intuition is that $A$ maps in one direction and $B$ in the other ($v$ and $w$ being balls crossing the "ping-pong table").
Now suppose you have two words in $A$ and $B$ that are equal, say
$$
ABBABA=AAABBB.
$$
Let's multiply by $A^{-1}$ and $B^{-1}$ as needed to strip off any leading identical letters; we get
$$
BBABA=AABBB.
$$
Well, then with $w=BABAx$ and $v=ABBBx$ (for any ping-pong ball $x$) we get $Av=Bw$, which was supposed to be impossible. Done!