Abstract homomorphisms of algebraic groups : rigidity and group actions

We investigate two related problems involving abstract homomorphisms between the groups of rational points of algebraic groups. First, we show that under appropriate assumptions, abstract representations of quasi-split special unitary groups associated with quadratic extensions of the field of definition have standard descriptions, i.e. can be factored as a group homomorphism induced by a morphism of algebras, followed by a homomorphism arising from a morphism of algebraic groups. This establishes a new case of a longstanding conjecture of Borel and Tits. In the second part, we apply existing results on standard descriptions for abstract representations of Chevalley groups to study some rigidity properties of actions of elementary subgroups on algebraic varieties. The thesis is organized as follows. To provide context for the study of abstract homomorphisms, in section 1 we give a historical overview of key developments going back to Cartan's work on homomorphisms of Lie groups. In section 2, we prove our rigidity result for special unitary groups, using a strategy inspired by work of Igor Rapinchuk which depends crucially on the construction of certain algebraic rings associated to abstract representations. In section 3, we apply existing rigidity statements for representations of elementary subgroups of Chevalley groups to study rigidity properties of these groups acting on affine algebraic varieties and projective surfaces. We discuss some open questions and plans for future work in section 4. In the appendices, we collect some relevant background material on algebraic rings, and also provide details on the computations of commutator relations needed for the constructions in section 2.