La paradoxa de Banach-Tarski i el semigrup de tipus

Authors

  • Pere Ara Bertran

Abstract

In this article we will study a key concept in relation to the well-known Banach-Tarski paradox. This is the concept of equidecomposability of subsets of a set X with respect to the action of a discrete group G. A subset E of X is G-paradoxical if there are two disjoint subsets E1 and E2 of E such that each of them is equidecomposable with E. The study of this relationship can be systematized by introducing a specific semigroup S (X, G), called the type semigroup of X. We will explain the use of the type semigroup S (X, G); in the proof of Tarski's Theorem. We will also introduce some generalizations of this concept to a topological setting, and we will consider the problem of the validity of Tarski's Theorem within this new context. In addition, we will review a recent result by Grabowski, Máthé and Pikhurko which gives a positive solution to the measurable circle squaring problem.

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Published

2020-09-08

How to Cite

Ara Bertran, P. (2020). La paradoxa de Banach-Tarski i el semigrup de tipus. Butlletí De La Societat Catalana De Matemàtiques, 35(1), 5–22. Retrieved from https://revistes.iec.cat/index.php/BSCM/article/view/107696.003

Issue

Vol. 35 No. 1 (2020)

Section

Articles

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