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A short proof of the Grigorchuk-Cohen cogrowth theorem

Author: Ryszard Szwarc
Journal: Proc. Amer. Math. Soc. 106 (1989), 663-665
MSC: Primary 43A07; Secondary 20E99, 20F05, 22D05
MathSciNet review: 975660
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Abstract: Let $ G$ be a group generated by $ {g_1}, \ldots ,{g_r}$. There are exactly $ 2r{(2r - 1)^{n - 1}}$ reduced words in $ {g_1}, \ldots ,{g_r}$ of length $ n$. Part of them, say $ {\gamma _n}$ represents identity element of $ G$. Let $ \gamma = \lim \sup \gamma _n^{1/n}$. We give a short proof of the theorem of Grigorchuk and Cohen which states that $ G$ is amenable if and only if $ \gamma = 2r - 12$. Moreover we derive some new properties of the generating function $ \sum {{\gamma _n}{z^n}} $.

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Article copyright: © Copyright 1989 American Mathematical Society

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