Date:
Thu, 19/12/201912:00-13:00
December 19, 12:00-13:00, Seminar room 209, Manchester building.
Abstract: 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 \hat{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 1 ≠ g ∈\hat{F}(p) that vanishes under every (continuous) homomorphism \hat{F}(p)→GL_(R) when R is a profinite commutative ring. In particular, when p>2 , \hat{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: Let n ∈ N . 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 probelm, 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 char(R)=2 (joint with E. Zelmanov).
Abstract: 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 \hat{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 1 ≠ g ∈\hat{F}(p) that vanishes under every (continuous) homomorphism \hat{F}(p)→GL_(R) when R is a profinite commutative ring. In particular, when p>2 , \hat{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: Let n ∈ N . 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 probelm, 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 char(R)=2 (joint with E. Zelmanov).