## The Tarski alternative and the Garden-of-Eden theorem

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