Pro-p identities of linear groups

It is a classical fact that free (discrete) groups can be embedded in GL_2(Z).  In 1987, Zubkov showed that for a non-abelian free pro-p group F^(p), the situation changes, and for every p > 2, groups of the form GL_2(R) satisfy a "pro-p identity."  More formally, for every p > 2 there exists a nontrivial element g of F^(p) that vanishes under every (continuous) homomorphism F^(p) --> GL_2(R), when R is a pro-finite commutative ring.  In particular, when p > 2, F^(p) cannot be embedded in GL_2(R).  

In 2005, inspired by the solution of the Specht problem, Zelmanov sketched a proof for the following generalization: if n is a natural number, then for every p >> n, GL_n(R) satisfies a "pro-p identity."

In the talk I will discuss Zelmanov's approach, its connection to the Specht problem, and its implications to the area of polynomial identities of Lie algebras.  In addition, I will discuss a recent result regarding the case p = n = 2, saying that GL_2(R) satisfies a pro-2 identity provided that char(R) = 2 (joint with E. Zelmanov).

See attached file.