The Growth of Grigorchuk's Group
  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 $\beta=\log_{32}(31)\approx0.991$. Using elementary methods we bring the upper bound down to $\log(2)/\log(2/\eta)\approx0.767$, where $\eta\approx0.811$ is the real root of the polynomial $X^3+X^2+X-2$. 
Laurent Bartholdi