The Tarski alternative and the Garden-of-Eden theorem

Silvio Capobianco

Institute of Cybernetics at TUT

Thursday, 3 May 2012, 14:00
Cybernetica Bldg (Akadeemia tee 21), room B101

Slides from the talk [pdf]

Abstract: The discovery of the Banach-Tarski paradox and the study of the axiomatic properties of the Lebesgue integral led to the development of a broad area of research joining measure theory, real analysis, and group theory. Amenable groups were defined in 1929 by John von Neumann as having a finitely additive probability measure which is invariant by left multiplication. They were later proved by Alfred Tarski to be all and only those that do not allow the kind of paradoxical decomposition which, in turn, causes the Banach-Tarski phenomenon. After giving the basic definitions, we will discuss the properties of amenable and paradoxical groups, and see how each group belongs to exactly one of the two classes. We will put special focus on a remarkable link with cellular automata theory: namely, the characterization, due to Laurent Bartholdi based on previous work by Tullio Ceccherini-Silberstein et al., of amenable groups as those where Moore's Garden-of-Eden theorem holds.

Tarmo Uustalu
Last update 8.5.2012