If G is a finitely generated group with vcd(G) finite, is vcd(H) finite for H, where H is an automorphism group of G?

Question

$\DeclareMathOperator\vcd{vcd}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\Out{Out}$Here I mean $\vcd(G)$ to be the virtual cohomological dimension of $G$. Some possible examples of what I mean by $H$ are $H=\Aut(G)$, $\Inn(G)$ or $\Out(G)$. I could extend this to other natural, more topological examples as well.

I've wondered about this and wondered if anyone actually speculated about this since the work in the 70s and 80s studying this problem where G is either a surface group (Harer, Penner, Thurston, etc.) or a free group (Culler, Vogtmann, Charney?, Bestvina, etc.). Of course we have the work of Borel, Serre and others for arithmetic groups, where G could be free abelian, or probably polycyclic-by-finite.

I would reframe and expand this question as follows: Let $G$ have $\vcd(G)$ finite so we have a finite dimensional model $X$ of $K(G,1)$. How can we construct a suitably good finite dimensional model $Y$ of $K(\Aut(G),1)$, for example?

As a side question, are there groups $G$ with $\vcd(G)$ finite but infinitely generated center $Z(G)$? What happens with $\Inn(G)=G/Z(G)$ then?

Answer 1

Sorry, it is Rips' and not Mikhailova's construction: I should not comment when I am half-asleep.

Let me start with the Rips construction.

Let $Q$ be a finitely presented group. Rips in [1] constructed $C'(1/\lambda)$-small cancellation groups $G$ (with arbibtarily large $\lambda$) and normal finitely generated subgroups $N< G$ such that $G/N\cong Q$. For $\lambda\ge 7$ the group $G$ will be hyperbolic.

A nice exposition of the Rips construction and its generalizations can be found in these two blog-posts: here and here. Actually, the Rips construction is quite flexible and one can make choices so that no defining relator of $G$ is a proper power; hence, the presentation complex of $G$ is aspherical. In particular, $G$ is torsion-free and is 2-dimensional. The subgroup $N$, therefore, is also 2-dimensional. However, the group $Q$ can be taken to have infinite virtual cohomological dimension.

We, thus, obtain a finitely generated 2-dimensional group $N$ such that $Out(N)$ contains $Q$ and thus has infinite vcd.

I am not sure how to find examples where $Aut$ has infinite vcd.

[1] Rips, E., Subgroups of small cancellation groups, Bull. Lond. Math. Soc. 14, 45-47 (1982). ZBL0481.20020.

Edit. A modification of the Rips construction in [2] yields examples where $Out(N)\cong Q$ even though it is not need for your question.

[2] Bumagin, Inna; Wise, Daniel T., Every group is an outer automorphism group of a finitely generated group., J. Pure Appl. Algebra 200, No. 1-2, 137-147 (2005). ZBL1082.20021.