The automorphism group of a free group is not linear (Q1194049)

The main result of this paper is that for \(n \geq 3\) the automorphism group of a free group of rank \(n\) is not a linear group -- i.e., it has no faithful representation by matrices over a field. The proof uses the representation theory of algebraic groups to show that the HNN-extension \({\mathcal H}(G) = \langle G \times G,t\mid t(g,g)t^{-1} = (1,g)\text{ for all }g\in G\rangle\) cannot be a linear group if \(G\) is not nilpotent-by- (abelian-by-finite). The main result is then proved by showing that for \(n\geq 3\) the automorphism group of a free group of rank \(n\) contains \({\mathcal H}(F_ 2)\), where \(F_ 2\) is a free group of rank two. The linearity of the automorphism group of a free group of rank two is an open question.