RANDOM WALKS ON FREE PERIODIC GROUPS

An upper estimate is obtained for the growth exponent of the set of all uncancellable words equal to 1 in a group given by a system of defining relations with the Dehn condition. By a theorem of Grigorchuk, this yields a sufficient test for the transience of a random walk on a group given by a system of defining relations with the Dehn condition, and for the nonamenability of such a group. It is proved that the free periodic groups B(m,n) with m?2 and odd n?665 satisfy this test. A question asked by Kesten in?1959 is thereby answered in the negative, and a conjecture put foth earlier by the author is confirmed. Bibliography: 7 titles.