On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group G we show that its growth $$v_{G,S}(n)$$ v G , S ( n ) satisfies $$\lim _{n\rightarrow \infty }\log \log v_{G,S}(n)/\log n=\alpha _{0}$$ lim n → ∞ log log v G , S ( n ) / log n = α 0 , where $$\alpha _{0}=\frac{\log 2}{\log \lambda _{0}}\approx 0.7674$$ α 0 = log 2 log λ 0 ≈ 0.7674 , $$\lambda _{0}$$ λ 0 is the positive root of the polynomial $$X^{3}-X^{2}-2X-4$$ X 3 - X 2 - 2 X - 4 .

中文翻译：

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周期性 Grigorchuk 群的增长

在扭力 Grigorchuk 群上，我们构造了有限熵和幂律尾衰减的随机游走，具有非平凡泊松边界。这种随机游走为这些组提供了接近最佳的体积较低的估计。特别地，对于第一个 Grigorchuk 组 G，我们证明其增长 $$v_{G,S}(n)$$ v G , S ( n ) 满足 $$\lim _{n\rightarrow \infty }\log \ log v_{G,S}(n)/\log n=\alpha _{0}$$ lim n → ∞ log log v G , S ( n ) / log n = α 0 ，其中 $$\alpha _{ 0}=\frac{\log 2}{\log \lambda _{0}}\approx 0.7674$$ α 0 = log 2 log λ 0 ≈ 0.7674 , $$\lambda _{0}$$ λ 0 是多项式 $$X^{3}-X^{2}-2X-4$$ X 3 - X 2 - 2 X - 4 的正根。