Profinite completion of Grigorchuk's group is not finitely presented

Abstract: In this paper we prove that the profinite completion $\mathcal{\hat G}$ of the Grigorchuk group $\mathcal{G}$ is not finitely presented as a profinite group. We obtain this result by showing that $H^2(\mathcal{\hat G},\field{F}2)$ is infinite dimensional. Also several results are proven about the finite quotients $\mathcal{G}/ St{\mathcal{G}}(n)$ including minimal presentations and Schur Multipliers.