| This thesis focuses primarily on the study of abstract representations of elementary subgroups of Chevalley groups over rings. Motivated by a conjecture of Borel and Tits on abstract homomorphisms of algebraic groups, we show that linear representations of elementary subgroups of Chevalley groups admit standard descriptions in many situations. The new approach to the conjecture that we develop in this thesis relies on several ingredients, including the study of algebraic rings and centrality results for K2. In fact, our methods yield rigidity results not only for Chevalley groups, but also for nonsplit groups of the form SLn,D (n ≥ 3), where D is a finite-dimensional central division algebra over a field of characteristic 0. As an application, we show how our results on abstract homomorphisms can be used to investigate deformations of representations of elementary subgroups of Chevalley groups over finitely generated commutative rings. Yet another approach to the study of abstract homomorphisms, which goes back to the work of Bass, Milnor, and Serre, relies on connections with the congruence subgroup problem; motivated by this, we establish a general centrality result for the congruence kernel of elementary subgroups of Chevalley groups. |
