Geodesic
Random walks on free periodic groups
Izvestiya. Mathematics , Tome 21 (1983) no. 3, pp. 425-434.

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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. Bibliography: 7 titles. 
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S. I. Adian. Random walks on free periodic groups. Izvestiya. Mathematics , Tome 21 (1983) no. 3, pp. 425-434. http://geodesic.mathdoc.fr/item/IM2_1983_21_3_a0/

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