The Nonfinite Generation of Aut(G), G Free metabelian of Rank 3

Abstract

The group of automorphisms of the free metabelian group of rank 3 is not finitely generated. Let H be a free solvable group of rank n > 2, Aut(H) the automorphism group of H, and Inn(H) the group of inner automorphisms of H. For n = 2, it was shown by the authors and E. Formanek [4, Theorem 1] that Aut(H)/Inn(H)GL2(Z), Z the ring of integers. One consequence of this fact is that Aut(H) is finitely generated (f.g.). The purpose of this paper is to prove the following contrasting theorem. THEOREM. If G is the free metabelian group of rank three, then Aut(G) is not finitely generated. This result may also be compared with a theorem of L. Auslander [1], which states that the automorphism group of a polycyclic group is finitely presented. If G' denotes the commutator subgroup of G, then the kernel of the natural homomorphism Aut(G) -4 Aut(G/G') is called the group of IA-automorphisms of G and will be denoted by IA(G). IA(G), which contains Inn(G), was shown by the authors [3] to be non-f.g. for G as in the Theorem. To prove that Aut(G) is not f.g., we need to show that IA(G) is not f.g. as an Aut(G/G') -GL3(Z)-operator group. The question arises concerning Aut(H) where H is free metabelian of rank n > 3. The immediate feeling that Aut(H) is also non-f.g. may be incorrect. For definiteness let n = 4. Suppose G is free metabelian of rank 3, and consider Aut(G) as embedded in Aut(H) in an obvious manner. The nonfinite generatiion of Aut(G) comes from the existence of automorphisms in Aut(G), i.e., automorphisms of G F/F which are not induced by automorphisms of the free group F of rank three, and the necessity to include infinitely many nontame automorphisms in any generating set for Aut(G). However, the authors have discovered that many of the nontame automorphisms become tame when considered as elements in Aut(H). Whether this phenomenon is true for all nontame elements of Aut(G) is unknown as yet, but the question of the finite or nonfinite generation of Aut(H) must be considered a difficult open problem, with the interesting possibility of finite generation as the answer. Received by the editors December 31, 1980. 1980 Mathematics Subject Classification. Primary 20F28.