Specht Problem and Gelfand Conjecture

Let $F$ be a free pro-$p$ non-abelian group, and let $\Delta$ be a commutative Noetherian complete local ring with maximal ideal $I$ such that $\mathrm{char}(\Delta/I)=p$.\\ We define the pro-$p$ group \[GL_2^1(\Delta) = \mathrm{ker}\left( GL_2(\Delta) \xrightarrow{\Delta\to\Delta/I} GL_2(\Delta/I) \right)\] A.N.\ Zubkov proved that $F$ cannot be continuously embedded in $GL_2^1(\Delta)$ for $p\neq 2$.\\ D.\ Ben-Ezra and E.\ Zelmanov further established that this embedding is not possible for $p = 2$ and $\mathrm{char}(\Delta) = 2$.\\ In this paper we aim to extend this result for $p=2$ and $\mathrm{char}(\Delta)=4$.\\ As was shown in~\cite{Zelmanov1}, the conjecture of non-linearity of free pro-$p$ groups is strongly connected with PI-theory. In the second part we will investigate the connection between PI-theory and the old-standing Gelfand conjecture. Thus, one can see that this paper is devoted to investigating combinatorial properties of substitutions.