EVENT DETAILS AND ABSTRACT

Title: On the conjecture of Borel and Tits for abstract homomorphisms of algebraic groups
Speaker: Igor Rapinchuk
Speaker Info: Harvard
Brief Description:
Special Note:
Abstract:

The conjecture of Borel-Tits (1973) states that if $G$ and $G'$ are algebraic groups defined over infinite fields $k$ and $k'$, respectively, with $G$ semisimple and simply connected, then given any abstract representation $\rho \colon G(k) \to G' (k')$ with Zariski-dense image, there exists a commutative finite-dimensional $k'$-algebra $B$ and a ring homomorphism $f \colon k \to B$ such that $\rho$ can essentially be written as a composition $\sigma \circ F$, where $F \colon G(k) \to G(B)$ is the homomorphism induced by $f$ and $\sigma \colon G(B) \to G'(k')$ is a morphism of algebraic groups. We prove this conjecture in the case that $G$ is either a universal Chevalley group of rank $\geq 2$ or the group $\mathbf{SL}_{n, D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic 0 and $n \geq 3$, and $k'$ is an algebraically closed field of characteristic 0. In fact, we show, more generally, that if $R$ is a commutative ring and $G$ is a universal Chevalley-Demazure group scheme of rank $ \geq 2$, then abstract representations over algebraically closed fields of characteristic 0 of the elementary subgroup $E(R) \subset G(R)$ have the expected description. We also describe some applications of these results to character varieties of finitely generated groups.