PARADOXAL DECOMPOSITIONS OF FREE BURNSIDE GROUPS

The Tarski number T(G) of a group G is defined as a measure of the com plexity of the possible paradoxical decompositions of G. A method is d eveloped for estimating the Tarski number of G in terms of a relative growth or of the spectral radius of an appropriate simple random walk on G. In particular we produce explicitely a function f(m) such that T (G) less than or equal to f(m)(alpha(H)) for G = F-m/H, with F-m a fre e group of rank m and alpha(H) the relative growth of H in F-m. This a nd some other inequalities imply the estimate 6 less than or equal to T(B(m, n)) less than or equal to 14 for a free Burnside group B(m, n) on m greater than or equal to 2 generators of odd exponent n greater t han or equal to 665. (C) Academie des Sciences/Elsevier, Paris.