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

Rapinchuk, Igor

doi:10.2140/ant.2013.7.1685

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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

DOI: 10.2140/ant.2013.7.1685

Abstract

In this paper, we continue our study, begun in an earlier paper, of abstract representations of elementary subgroups of Chevalley groups of rank $\geq 2$ . 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 ${S L}_{n, D}$ , 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 $\geq 2$ 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

Vol. 7, No. 7, 2013