Profinite completion of Grigorchuk's group is not finitely presented

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

Quadratic equations over free groups are NP-complete

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete