The many faces of $E_0$, delivered at Boise State University, September 10. 2012.
This talk was originally announced here. It was a discussion of several occurrences of the equivalence relation $E_0$ in mathematics. Of course, $E_0$ is my favorite equivalence relation.
The first occurrence is in a logic puzzle about guessing the color of your hat. If this sounds all too familiar, well, there are many such puzzles. This version is an infinitary one, and was first considered by Lubarsky and Geschke. You can also find an awesome post about the problem by Richard Elwes here. (Thanks Andres!)
The second occurrence is in finding a Lebesgue nonmeasurable set.
The third occurrence is nearest and dearest to my own area of research: classifying the torsion-free abelian groups of rank 1.
You can see my notes here.
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In this context, what does “rank 1″ mean? An abelian group whose tensor product with $\mathbb{Q}$ is a 1-dimensional $\mathbb{Q}$-vector space? (Guessing.)
That is exactly right!
A simple way to think about it is the largest size of a subset of $A$ that is independent over $\mathbb Z$.
And yes, that is equivalent to say $A\otimes\mathbb Q\cong\mathbb Q$. Thus the rank 1 (such) groups are just the subgroups of $\mathbb Q$.