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

TL;DR

This paper advances the understanding of abstract representations of elementary subgroups of Chevalley groups over various rings, proving a key conjecture and exploring deformations of these representations.

Contribution

It extends methods to analyze representations over arbitrary rings and proves the Borel-Tits conjecture for certain algebraic groups, also studying their deformations.

Findings

Proved the Borel-Tits conjecture for groups of the form ${\bf SL}_{n,D}$.

Extended analysis methods to arbitrary associative rings.

Studied deformations of representations over finitely generated rings.

Abstract

In this paper, we continue our study of abstract representations of elementary subgroups of Chevalley groups of rank $\geq 2.$ First, we extend our earlier 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 ${\bf SL}_{n,D}$, where $D$ is a finite-dimensional central division algebra over a field of characteristic zero. Second, we apply our results to study deformations of representations of elementary subgroups of universal Chevalley groups of rank $\geq 2$ over finitely generated commutative rings.