The Growth of Grigorchuk's Group

Abstract

In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth function: $$e^{\sqrt{n}}\precsim \gamma (n)\precsim e^{n^{\beta }},$$ where $\beta ={\log }_{32}(31)\approx 0.991$. Using elementary methods we bring the upper bound down to $\log (2)/\log (2/\eta )\approx 0.767$, where $\eta \approx 0.811$ is the real root of the polynomial $X^3+X^2+X-2$.

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