On Pro-$2$ Identities of $2\times 2$ Linear Groups

Abstract

Let $\hat{F}$ be a free pro-$p$ non-abelian group, and let $\Delta$ be a commutative Noetherian complete local ring with a maximal ideal $I$ such that $\text{char}(\Delta /I)=p>0$. In [Zu], Zubkov showed that when $p\ne 2$, the pro-$p$ congruence subgroup $$G{L}_2^1(\Delta )=\ker (G{L}_2(\Delta )\stackrel{\Delta \to \Delta /I}{⟶}G{L}_2(\Delta /I))$$ admits a pro-$p$ identity, i.e., there exists an element $1\ne w\in \hat{F}$ that vanishes under any continuous homomorphism $\hat{F}\to G{L}_2^1(\Delta )$. In this paper we investigate the case $p=2$. The main result is that when $\text{char}(\Delta )=2$, the pro-$2$ group $G{L}_2^1(\Delta )$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that originate in PI-theory.

40 pages