The winding invariant

Abstract

Every element $w$ in the commutator subgroup of the free group ${\mathbb{F}}_2$ of rank 2 determines a closed curve in the grid $\mathbb{Z}\times \mathbb{R}\cup \mathbb{R}\times \mathbb{Z}\subseteq {\mathbb{R}}^2$. The winding numbers of this curve around the centers of the squares in the grid are the coefficients of a Laurent polynomial $P_w$ in two variables. This basic definition is related to well-known ideas in combinatorial group theory. We use this invariant to study equations over ${\mathbb{F}}_2$ and over the free metabelian group of rank $2$. We give a number of applications of algebraic, geometric and combinatorial flavor.

60 pages, 19 figures. Typos and a reference corrected

References in corpus (4)

• Word equations in simple groups and polynomial equations in simple algebras
• On the Surjectivity of Engel Words on PSL(2,q)
• Exotic relation modules and homotopy types for certain 1-relator groups
• A counterexample to a strong version of the Andrews-Curtis conjecture

Cited by in corpus (1)

• A counterexample to a strong version of the Andrews-Curtis conjecture