Geodesic
Random walks on free periodic groups
Izvestiya. Mathematics, Tome 21 (1983) no. 3, pp. 425-434 
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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     author = {S. I. Adian},
     title = {Random walks on free periodic groups},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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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[2] Adyan S. I., Problema Bernsaida i tozhdestva v gruppakh, Nauka, M., 1975 | MR | Zbl

[3] Grigorchuk R. I., “Simmetrichnye sluchainye bluzhdaniya na diskretnykh gruppakh”, Mnogokomponentnye sluchainye sistemy, Nauka, M., 1978, 132–152 | MR

[4] Kesten H., “Symmetric random walks on groups”, Trans. Amer. Math. Soc., 2 (1959), 336–354 | DOI | MR

[5] Kesten H., “Full Banach mean values on countable groups”, Math. scand., 7:1 (1959), 146–156 | MR | Zbl

[6] Adyan S. I., “Aksiomaticheskii metod postroeniya grupp s zadannymi svoistvami”, Uspekhi matem. nauk, 32:1 (1977), 3–15 | MR | Zbl

[7] Olshanskii A. Yu., “K voprosu o suschestvovanii invariantnogo srednego na gruppe”, Uspekhi matem. nauk, 35:4 (1980), 199–200 | MR