On pro-$2$ identities of $2\times 2$ linear groups

Let F ^ \hat {F} be a free pro- p encoding="application/x-tex">p non-abelian group, and let alttext="normal Delta"> Δ encoding="application/x-tex">\Delta commutative Noetherian complete local ring with maximal ideal alttext="upper I"> I encoding="application/x-tex">I such that c normal h r left-parenthesis Delta slash I right-parenthesis equals p greater-than 0"> mathvariant="normal">c mathvariant="normal">h mathvariant="normal">a mathvariant="normal">r stretchy="false">( / stretchy="false">) = &gt; 0 encoding="application/x-tex">\mathrm {char}(\Delta /I)=p&gt;0 . Zubkov [Sibirsk. Mat. Zh. 28 (1987), pp. 64–69] showed when alttext="p not-equals 2"> ≠<!-- ≠ <mml:mn>2 encoding="application/x-tex">p\neq 2 , the congruence subgroup G L 1 ker ⁡<!-- ⁡ <mml:msub> stretchy="false">⟶<!-- ⟶ stretchy="false">→<!-- → encoding="application/x-tex">GL_{2}^{1}(\Delta )=\ker (GL_{2}(\Delta )\overset {\Delta \to \Delta /I}{\longrightarrow }GL_{2}(\Delta /I)) \] admits identity, i.e., there exists an element alttext="1 w element-of ModifyingAbove w ∈<!-- ∈ encoding="application/x-tex">1\neq w\in \hat vanishes under any continuous homomorphism caret {F}\to GL_{2}^{1}(\Delta ) In this paper we investigate case encoding="application/x-tex">p=2 The main result is )=2 alttext="2"> encoding="application/x-tex">2 group identity. This was obtained by use of trace identities originate in PI-theory.