Cohomological dimension of direct product

Question

Let $\operatorname{cd}$ denote the cohomological dimension of a group, i.e. the minimal length of a projective resolution of $\mathbb{Z}$ over the group ring.

Let $G_1$ and $G_2$ be groups. It is easy to see that $$\operatorname{cd}(G_1 \times G_2) \leq \operatorname{cd}(G_1)+\operatorname{cd}(G_2) \ , $$ (using the tensor product of resolutions), but is there a clear criterion that implies equality? In particular, are there groups $G$ with $\operatorname{cd}(G\times G) < 2 \operatorname{cd}(G)$?

Answer 1

There is a criterion for equality in a more general setup. Given a short exact sequence of groups

$$ 1 \to N \to G \to Q \to 1 $$

where $\text{cd}(Q)<\infty$, the group $N$ is of type FP (meaning $\mathbb{Z}$ admits a finite length resolution by finitely generated $\mathbb{Z}N$-projective modules) and $H^n(N;\mathbb{Z}N)$ is free for $n=\text{cd}(N)$, then

$$ \text{cd}(G) = \text{cd}(N) + \text{cd}(Q) $$

This is Theorem 5.5 in the book "Homological Dimension of Discrete Groups" by R. Bieri.

I do not know any example of a group with $\text{cd}(G \times G) < 2\text{cd}(G)$, but there is an example in the book "The geometry and topology of Coxeter groups" by M. W. Davis of groups $G_1$ and $G_2$ with $\text{cd}(G_1 \times G_2) < \text{cd}(G_1) + \text{cd}(G_2)$. It is example 8.5.9 in page 157.

In fact, this example describes two Coxeter groups $W'$ and $W''$ such that

$$ \text{vcd}(W' \times W'') < \text{vcd}(W') + \text{vcd}(W'') $$

Since the virtual cohomological dimension is the cohomological dimension of any torsion-free subgroup of finite index, the corresponding subgroups $G_1$ of $W'$ and $G_2$ of $W''$ satisfy the strict inequality.

This example is attributed to the article "On the Virtual Cohomological Dimensions of Coxeter Groups" by A.N. Dranishnikov. In this article Corollary 1 says that for a Coxeter group $\Gamma$ we always have $\text{vcd}(\Gamma \times \Gamma) = 2\text{vcd}(\Gamma)$, so you have $\text{cd}(G \times G) = 2\text{cd}(G)$ for any torsion-free subgroup of finite index of a Coxeter group.

Answer 2

Let $A$ and $B$ be infinitely generated subgroups of the additive group of rational numbers. Then $cd(A) = cd(B) = 2$ and $cd(A \times B) = 3$. In particular, for such an $A$, $$cd(A \times A) < 2.cd(A).$$

More generally this pathology occurs if one takes $A$ and $B$ to be any soluble groups of finite cohomological dimension which are not of type $F$. Then $cd(A \times B)$ is always exactly one less than $cd(A) + cd(B)$.

Answer 3

There is a recent theorem* which gives a clear criterion for equality:

Theorem. If $G$ has a finite Eilenberg-MacLane space (a finite $K(G, 1)$) then $\operatorname{cd}(G\times G) = 2 \operatorname{cd}(G)$.

Therefore, any group $G$ with $\operatorname{cd}(G\times G) < 2 \operatorname{cd}(G)$ cannot be geometrically finite. Note that $\mathbb{Q}$ is not geometrically finite, and so this does not contradict Peter Kropholler's example of $\operatorname{cd}(\mathbb{Q\times Q}) < 2 \operatorname{cd}(\mathbb{Q})$.

*Dranishnikov, Alexander. "On dimension of product of groups." Algebra and Discrete Mathematics 28.2 (2020). (arXiv)