## Modular Makeup

The Axiom of Choice can be a troublesome beast. It leads, for instance, to the amazing Banach-Tarski paradox, which states that one can cut up an orange and use the pieces to make two oranges. I found a wonderful book in the library on this topic, by Stan Wagon (here is the Google peek page).

A subset \$U\$ of a set \$X\$ is said to be paradoxical with respect to a group \$G\$ (used to rearrange pieces) if such a process can be done to \$U\$. The orange example comes from considering balls in \$\mathbb{R}^{3}\$ and translations and rotations, and it was shown by R. Robinson that only five pieces are needed to make a paradoxical orange!

Using the upper half plane and the modular group one can study similar paradoxes using Borel sets. Hausdorff showed, using the \$S\$ and \$T\$ presentation embedded in a rotation group, that the modular group is paradoxical. The relevant decomposition of hyperbolic space is three pieces \$A\$, \$B\$ and \$C\$ (see the pretty picture on the book cover) which are related via \$TA = B\$, \$T^{2} A = C\$ and \$S A = B \cup C\$.

What Tarski showed was that paradoxical decompositions are really about the non-existence of a finitely additive invariant measure.

## 4 Responses so far »

1. 1

### CarlBrannen said,

Marni, I threw \$15 at wordpress and redid the format for my .

2. 2

### CarlBrannen said,

Dang! That looked fine in “preview”. You should dump blogger and move over to wordpress.

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### Matti Pitkanen said,

It is interesting that the congruence of open sets with non-empty interiors holds true only for D>2. In plane results are much weaker and area is same for congruent sets. Brings in mind 2-dimensionality of partonic 2-surfaces.

I would guess that the paradox disapperas if one can replace the axiom of choice with a weaker form restricting the choice to rationals or algebraics. This kind
of restriction of course makes sense only in a very special case when one has symmetries so that one can specify the preferred coordinates with respect to which the notion of rational point is defined.

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### Kea said,

Cool, Carl! I’ve made it my main link to you on my blogroll now. I’ve put some work into blogger and am a little reluctant to redo everything again at the moment. Maybe if I get a life one day.

I would guess that the paradox disappears if one can replace the axiom of choice …

Hi Matti. At the end of the book, Wagon discusses the close relation between AC and the paradox: one can show that the paradox is unprovable in ZF alone, so it really hinges on AC. In the topos Set, AC is a simple condition but not one of the elementary axioms, so it is easy enough in higher topos theory to work without AC. I like this idea for many reasons.