quality:corruption, structural:scholar_edition, linguistic:translation
Step 1

The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a, b and b such that no a appears directly next to an a and no b appears directly next to a b. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: abab a concatenated with abab a yields abab a abab a, which contains the substring a a, and so gets reduced to abab bab a, which contains the substring b b, which gets reduced to abaab a. One can check that the set of those strings with this operation forms a group with identity element the empty string e. This group may be called F.

The group F 2 {\displaystyle F_{2}} can be "paradoxically decomposed" as follows: Let S (a) be the subset of F 2 {\displaystyle F_{2}} consisting of all strings that start with a, and define S (a), S (b) and S (b) similarly. Clearly,

but also

and

where the notation aS (a) means take all the strings in S (a) and concatenate them on the left with a.

This is at the core of the proof. For example, there may be a string a a − 1 b {\displaystyle aa^{-1}b} in the set a S (a − 1) {\displaystyle aS(a^{-1})} which, because of the rule that a {\displaystyle a} must not appear next to a − 1 {\displaystyle a^{-1}}, reduces to the string b {\displaystyle b}. Similarly, a S (a − 1) {\displaystyle aS(a^{-1})} contains all the strings that start with a − 1 {\displaystyle a^{-1}} (for example, the string a a − 1 a − 1 {\displaystyle aa^{-1}a^{-1}} which reduces to a − 1 {\displaystyle a^{-1}}). In this way, a S (a − 1) {\displaystyle aS(a^{-1})} contains all the strings that start with b {\displaystyle b}, b − 1 {\displaystyle b^{-1}} and a − 1 {\displaystyle a^{-1}}, as well as the empty string e {\displaystyle e}.

Group F has been cut into four pieces (plus the singleton { e }), then two of them "shifted" by multiplying with a or b, then "reassembled" as two pieces to make one copy of F 2 {\displaystyle F_{2}} and the other two to make another copy of F 2 {\displaystyle F_{2}}. That is exactly what is intended to do to the ball.