On pro-2 identities of 2×2 linear groups

Ben-Ezra, David El-Chai; Zelmanov, Efim

doi:10.1090/tran/8327

On pro-$2$ identities of $2\times 2$ linear groups
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Trans. Amer. Math. Soc. 374 (2021), 4093-4128 Request permission

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 $\mathrm {char}(\Delta /I)=p>0$. Zubkov [Sibirsk. Mat. Zh. 28 (1987), pp. 64–69] showed that when $p\neq 2$, 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 $\mathrm {char}(\Delta )=2$, 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.

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