Texas A&M University, Department of
Mathematics, 216 Milner Hall, 6th of October 2004, 3:00-3:50
Groups and Dynamics Seminar
Amenability and
paradoxical decompositions: Tarski's alternative theorem
Tullio Ceccherini-Silberstein of
Universitá del Sannio, Benevento, Italy
We present a proof, obtained in collaboration with Pierre de la Harpe
and Slava Grigorchuk, of a theorem of Tarski which asserts that a
(discrete) group is either amenable (i.e. it bears a left-invariant
finitely additive proabability measure) or it is paradoxical (in the
sense of the Haussdorff-Banach-Tarski paradox). The techniques come
from functional analysis and probability (random walks on discrete
groups, spectral radius) and combinatorics (Rado-Hall marriage theorem
on matchings). As a bypass of our methods we obtain estimates for the
Tarski number (the minimal number of "pieces" involved in a paradoxical
decomposition) of the free Burnside groups B(m,n), for odd n > 665
and any m > 1.