Profinite completion of Grigorchuk's group is not finitely presented

TL;DR

This paper proves that the profinite completion of the Grigorchuk group is not finitely presented by demonstrating the infinite dimensionality of its second cohomology group, and also explores properties of its finite quotients.

Contribution

It establishes the non-finite presentability of the profinite completion of the Grigorchuk group and analyzes related finite quotients and their properties.

Findings

Profinite completion of Grigorchuk's group is not finitely presented.

Second cohomology group of the profinite completion is infinite dimensional.

Finite quotients have minimal presentations and known Schur multipliers.

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.