Construction of a quotient ring of ${\mathbb{Z}}_2\mathcal{F}$ in which a binomial $1+w$ is invertible using small cancellation methods

Abstract

We apply small cancellation methods originating from group theory to investigate the structure of a quotient ring ${\mathbb{Z}}_2\mathcal{F}/\mathcal{I}$, where ${\mathbb{Z}}_2\mathcal{F}$ is the group algebra of the free group $\mathcal{F}$ over the field ${\mathbb{Z}}_2$, and the ideal $\mathcal{I}$ is generated by a single trinomial $1+v+vw$, where $v$ is a complicated word depending on $w$. In ${\mathbb{Z}}_2\mathcal{F}/\mathcal{I}$ we have $(1+w{)}^{-1}=v$, so $1+w$ becomes invertible. We construct an explicit linear basis of ${\mathbb{Z}}_2\mathcal{F}/\mathcal{I}$ (thus showing that ${\mathbb{Z}}_2\mathcal{F}/\mathcal{I}\ne 0$). This is the first step in constructing rings with exotic properties.

To be published in Contemporary Mathematics, Israel Mathematical Conference Proceedings (IMCP), 2019 Reference to a grant is added