Quadratic equations in the Grigorchuk group

TL;DR

This paper presents an algorithm to solve quadratic equations in the Grigorchuk group and proves that the group has finite commutator width, advancing understanding of its algebraic structure.

Contribution

The paper introduces a novel algorithm for solving quadratic equations in the Grigorchuk group and establishes finite commutator width as a new property.

Findings

Algorithm determines solvability of quadratic equations in the Grigorchuk group

Proof that the Grigorchuk group has finite commutator width

Enhanced understanding of the group's algebraic properties

Abstract

We provide an algorithm which, for a given quadratic equation in the Grigorchuk group determines if it has a solution. As a corollary to our approach, we prove that the group has a finite commutator width.