John von Neumann introduced the notion of amenable groups while studying the algebraic properties that make this result possible. For 10 points each:
[10m] Name this counterintuitive result which states that the group action of SO(3) admits a “paradoxical decomposition” into disjoint pieces that may be rearranged to form two copies of the original set.
ANSWER: Banach–Tarski paradox
[10h] Based on the Banach–Tarski proof, Von Neumann falsely conjectured that all non-amenable groups contain this group as a subgroup. This is the fundamental group of a singly punctured torus, or S-one wedge S-one.
ANSWER: free group on 2 generators [or F2 or F-sub-2 or the free group of rank 2; accept Z free Z or Z star Z; accept anything reasonable mentioning both free and 2, or both free and two copies of the integers; prompt on free group; reject “Z cross Z” or “Z times Z”]
[10e] The Banach–Tarski paradox concerns decompositions of these sets consisting of all points in space that are a fixed distance from a center point.
ANSWER: 2-spheres [or S-2; accept unit spheres or n-spheres or S-n]
<Jeremy Cummings, Other Science>