# Groups whose automorphism tower is finitely generated

Let $$G$$ be a finitely generated group. Then the automorphism group $$\text{Aut}(G)$$ of $$G$$ need not be finitely generated.

However, there are classes of f.g. groups for which the automorphism group will always be f.g., such as polycyclic groups: it is a result by Auslander from 1969 that the automorphism group of a polycyclic group is even finitely presented.

Let $$\text{Aut}^0(G) := G$$, and for $$n \geq 1$$ let $$\text{Aut}^n(G)$$ be defined inductively as $$\text{Aut}(\text{Aut}^{n-1}(G))$$.

My question is: what are some examples of groups $$G$$ such that $$\text{Aut}^n(G)$$ is f.g. for all $$n \geq 0$$?

Now, while $$\text{Aut}(G)$$ is f.g. if $$G$$ is polycyclic by the above, the automorphism group of a polycyclic group need not be polycyclic, as far as I am aware, which would indicate that it is at least conceivable that $$\text{Aut}^n(G)$$ need not be f.g. for all $$n \geq 1$$ when $$G$$ is polycyclic. But I do not know of any counterexamples in this class.

Note that any finite group satisfies the question. Also, if $$G$$ is taken as $$\mathbb{Z}$$, then its automorphism group is $$C_2$$. Are there any infinite examples where every automorphism group is infinite f.g.?

• $\text{Aut}(\Bbb Z)=C_2$ and from there the tower is made of finite groups. Nov 6, 2019 at 9:25
• @ArnaudMortier I was just writing an edit to address this! Thanks. Nov 6, 2019 at 9:26
• The automorphism group of a free group of finite rank is finitely generated. Nov 6, 2019 at 16:54
• @DerekHolt Yes, of course you're right -- even finitely presented. I had in mind a Baumslag-Solitar group as a counterexample, and free groups of finite rank as an example. Somehow they got swapped around in my mind when writing this up... Nov 7, 2019 at 0:42

If $$G$$ is a complete group, that is, with trivial center and no outer automorphisms, then $$G\simeq \text{Aut}(G)$$ as every automorphism is the conjugation by some element, and the map $$g\mapsto g\cdot g^{-1}$$ has a trivial kernel.
• Guba's examples have properties much harder to obtain (uniquely divisible). A low-tech example of a complete group, in addition finitely presented, is $\mathrm{GL}_d(\mathbf{Z})\ltimes\mathbf{Z}^d$ for $d\ge 3$.