Matrix entries, unipotents, and linearity of amalgams

TL;DR

This paper studies the linearity properties of amalgamated subgroups within algebraic groups, introduces new linearity results for certain doubled groups, and provides examples of finitely generated residually finite groups that are not linear.

Contribution

It establishes linearity of specific doubled linear groups and presents new examples of finitely generated residually finite groups that are non-linear.

Findings

Linearity of certain doubles of linear groups.

Existence of finitely generated residually finite non-linear groups.

Insights into the structure of amalgams in algebraic groups.

Abstract

We investigate linearity of amalgams of subgroups of algebraic groups along intersections with algebraic subgroups. In the process, we establish linearity of certain "doubles" of linear groups, and obtain new examples of finitely generated residually finite groups that fail to be linear.