Growth of periodic Grigorchuk groups
Abstract
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 volume growth function $v_{G,S}(n)$ satisfies that $\lim_{n\to\infty}\log\log v_{G,S}(n)/\log n=\alpha_{0}$, where $\alpha_{0}=\frac{\log2}{\log\lambda_{0}}\approx0.7674$, $\lambda_{0}$ is the positive root of the polynomial $X^{3}-X^{2}-2X-4$.
- Publication:
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arXiv e-prints
- Pub Date:
- February 2018
- DOI:
- 10.48550/arXiv.1802.09077
- arXiv:
- arXiv:1802.09077
- Bibcode:
- 2018arXiv180209077E
- Keywords:
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- Mathematics - Group Theory;
- Mathematics - Probability