Abstract:
In 1924, Banach and Tarski showed that a solid ball in $\mathbb{R}^3$ can be broken into finitely many pieces which, in turn, can be used as a puzzle to form two balls identical to the original one. This result is so striking that it bacame known as the Banach-Tarski paradox, despite being a theorem, proven with full mathematical rigor. Tarski later found a reciprocal of this result, proving that, under the action of a group of symmetries, the only obstruction to the existence of an invariant measure is the presence of paradoxical sets, namely those which can be doubled in size, as the ball in $\mathbb{R}^3$. In this lecture I plan to take the ideas of Banach and Tarski into the realm of totally disconnected topological spaces, by restricting the notion of measure to the boolean algebra of clopen (closed and open) subsets. We will see that the theory of partial group actions provides an example forbidding the generalization of Tarski’s Theorem to this context. This example was found in collaboration with Pere Ara, from the Universitat Aut`onoma de Barcelona.