On abstract representations of the groups of rational points of algebraic groups and their deformations

Abstract

In this paper, we continue our study, begun in an earlier paper, of abstract representations of elementary subgroups of Chevalley groups of rank $\ge 2$. First, we extend the methods to analyze representations of elementary groups over arbitrary associative rings and, as a consequence, prove the conjecture of Borel and Tits on abstract homomorphisms of the groups of rational points of algebraic groups for groups of the form ${SL}_{n,D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic $0$. Second, we apply the previous results to study deformations of representations of elementary subgroups of universal Chevalley groups of rank $\ge 2$ over finitely generated commutative rings.