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

TL;DR

This paper proves that the pro-2 congruence subgroup of $GL_2( ext{complete local ring})$ admits a pro-2 identity, extending previous results known for odd primes using trace identities from PI-theory.

Contribution

It establishes the existence of a pro-2 identity for $GL_2^1( ext{ring})$ when the ring's characteristic is 2, filling a gap in the understanding of pro-$p$ identities.

Findings

Pro-2 group $GL_2^1( ext{ring})$ admits a pro-2 identity when $ ext{char}( ext{ring})=2$.

Extension of Zubkov's result from $p eq 2$ to $p=2$ case.

Use of trace identities from PI-theory to prove the existence of pro-$p$ identities.

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 $\textrm{char}(\Delta/I)=p>0$. 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.