Lower Bounds on the Growth of Grigorchuk's Torsion Group

Abstract

In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth $\gamma$: $e^{\sqrt{n}}\precsim \gamma (n)\precsim e^{n^{\beta }},$ where $\beta ={\log }_{32}(31)\approx 0.991$. He conjectured that the lower bound is actually tight. In this paper we improve the lower bound to $e^{n^{\alpha }}\precsim \gamma (n),$ where $\alpha \approx 0.5157$, with the aid of a computer. This disproves the conjecture that the lower bound be tight.

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