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 ${\alpha }_0=\frac{\log 2}{\log {\lambda }_0}\approx 0.7674$, ${\lambda }_0$ is the positive root of the polynomial $X^3-X^2-2X-4$.

Cited by in corpus (1)

• On Growth of Generalized Grigorchuk's Overgroups