Vol. 7, No. 7, 2013

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

Igor A. Rapinchuk

Vol. 7 (2013), No. 7, 1685–1723
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.

Keywords
abstract homomorphisms, algebraic groups, rigidity, character varieties
Primary: 20G15
Secondary: 14L15