Vol. 7, No. 7, 2013

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ISSN: 1944-7833 (e-only)
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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 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 SLn,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 2 over finitely generated commutative rings.

Keywords
abstract homomorphisms, algebraic groups, rigidity, character varieties
Mathematical Subject Classification 2010
Primary: 20G15
Secondary: 14L15
Milestones
Received: 9 June 2012
Revised: 15 June 2012
Accepted: 7 September 2012
Published: 12 October 2013
Authors
Igor A. Rapinchuk
Department of Mathematics
Harvard University
1 Oxford Street
Cambridge, MA
02138
United States