On Pro-$2$ Identities of $2\times2$ Linear Groups

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 $\textrm{char}(\Delta/I)=p>0$. In [Zu], Zubkov showed that when $p\neq2$, the pro-$p$ congruence subgroup $$GL_{2}^{1}(\Delta)=\ker(GL_{2}(\Delta)\overset{\Delta\to\Delta/I}{\longrightarrow}GL_{2}(\Delta/I))$$ admits a pro-$p$ identity, i.e., there exists an element $1\neq w\in\hat{F}$ that vanishes under any continuous homomorphism $\hat{F}\to GL_{2}^{1}(\Delta)$. In this paper we investigate the case $p=2$. The main result is that when $\textrm{char}(\Delta)=2$, the pro-$2$ group $GL_{2}^{1}(\Delta)$ admits a pro-$2$ identity. This result was obtained by the use of trace identities that originate in PI-theory.