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The automorphism group of a free group is not linear. (English) Zbl 0780.20023
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.

20F28 Automorphism groups of groups
20E05 Free nonabelian groups
20E07 Subgroup theorems; subgroup growth
20E36 Automorphisms of infinite groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Full Text: DOI
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