Linear automorphism groups of relatively free groups

TL;DR

This paper characterizes when the automorphism group of a relatively free group is linear, showing it occurs precisely when the group is virtually nilpotent, thus linking algebraic structure to linearity.

Contribution

It provides a complete characterization of linear automorphism groups for relatively free groups that are not absolutely free.

Findings

Aut(G) is linear iff G is virtually nilpotent

Relatively free groups have non-linear automorphism groups unless virtually nilpotent

The result connects group structure with linearity of automorphism groups

Abstract

Let G be a free group in a variety of groups, but G is not absolutely free. We prove that the group of automorphisms Aut(G) is linear iff G is a virtually nilpotent group.