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 $\mathbf B(m,n)$ with $m\geqslant2$ and odd $n\geqslant665$ 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.
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