Recently I've been reading about cohomological finiteness conditions for groups, my main source being Brown's book "Cohomology of Groups".

One of the first things one learns is that a group with finite cohomological dimension necessarily is torsionfree. It is easy to come up with an example of a torsionfree group with infinite cohomological dimension: non-finitely generated free abelian group immediately springs to mind. A little bit of googling revealed an example of an infinite-dimensional torsionfree $FP_{\infty}$ group.$^1$ (A group is said to be of type $FP_{\infty}$ if there exists a projective $ZG$-resolution {$P_i$} of $Z$ such that each $P_i$ is finitely generated.) This example is Thompson's group $F$. However, it is well-known that $F$ contains a non-finitely generated free abelian group.

So, the question is: what is an example of a torsionfree group with infinite cohomological dimension and no inifinitely generated free abelian subgroup (that is, if there exists one)?

$^1$ In case anyone is interested: K. S. Brown, R. Geoghegan, Invent. Math. 77, 367--381.


You can take a union of torsion-free word-hyperbolic groups with cohomological dimension approaching infinity to get such an example. Word-hyperbolic groups are rank one, so one can show that any abelian subgroup must intersect each subgroup in a cyclic group, and therefore the subgroup cannot be an infinitely generated free abelian subgroup.

Here's an explicit sequence. Take a sequence of cocompact torsion-free lattices in $\Gamma_n< Isom(\mathbb{H}^n)$, such that $\Gamma_n < \Gamma_{n+1}$ from the natural embedding $Isom(\mathbb{H}^n) < Isom(\mathbb{H}^{n+1})$. The limit $\Gamma_{\infty}=\underset{n\to\infty}{\lim} \Gamma_n$ is a group of cohomological dimension $\infty$ since $\Gamma_n$ is of cohomological dimension $n$.

For example, consider the quadratic form over $\mathbb{Q}(\sqrt{5})$ which is $q_n(x_0,x_1,\ldots,x_n)=((1-\sqrt{5})/2 )x_0^2+x_1^2+\cdots+x_n^2$. This gives a quadratic form over $\mathbb{R}$ of signature $(n,1)$. Consider the group $O(q_n,\mathbb{Z}[\phi])< GL(n+1,\mathbb{Q}(\sqrt{5}))$, where $\phi=(1+\sqrt{5})/2$, the group of integral matrices preserving this quadratic form. Then there exists a prime ideal $\mathcal{P}<\mathbb{Z}[\phi]$ such that $\Lambda_n=Ker\{ GL(n+1,\mathbb{Z}[\phi])\to GL(n+1,\mathbb{Z}[\phi]/\mathcal{P})\}$ is torsion-free for all $n$. Let $\Gamma_n=O(q_n,\mathbb{Z}[\phi])\cap \Lambda_n$. Then $\Gamma_n$ is a torsion-free hyperbolic group acting cocompactly on $\mathbb{H}^n$ (by considering the Lorentzian model of hyperbolic space), and clearly $\Gamma_n<\Gamma_{n+1}$.

  • 1
    $\begingroup$ Here is another way to get an increasing sequence of hyperbolic groups $G_n$ of cohomological dimension $n$: using strict relative hyperbolization one can show that any closed aspherical $n$-manifold with hyperbolic fundamental group is a retract of a closed aspherical $(n+1)$-manifold with hyperbolic fundamental group, see arxiv.org/abs/math/0509504 Now as in your argument the union of $G_n$'s is the desired example. I wonder if there is an $FP_\infty$ example. $\endgroup$ Apr 4 '11 at 17:47
  • $\begingroup$ @ Igor: I imagine the Higman embedding theorem should allow one to embed these examples into a finitely presented group that has the same property (no infinite rank abelian subgroup). However, I'm not sure this will be $FP_{\infty}$. $\endgroup$
    – Ian Agol
    Apr 4 '11 at 18:47

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