14.42 (1999)

Is a free pro-$p$-group representable by matrices over an associative-commutative profinite ring with 1?

A negative answer is equivalent to the fact that every linear pro-$p$-group satisfies a non-trivial pro-$p$-identity. This is known to be true in dimension 2 for $p \neq 2$ (A. N. Zubkov, Siberian Math. J., 28, no. 5 (1987), 742–747). It is also proved that a 2-dimensional linear pro-2-group in characteristic 2 satisfies a non-trivial pro-2-identity (D. E.-C. Ben-Ezra, E. Zelmanov, Trans. Amer. Math. Soc., 374, no. 6 (2021), 4093–4128).