In this recent preprint, the authors construct a certain uncountable family of non-finitely presented FP groups. Recall that group is an FP group if the trivial $\mathbb Z[G]$-module $\mathbb Z$ has a finite projective resolution by finitely generated $\mathbb Z[G]$-projectives.

In fact, if I understand correctly, the family that they construct is a family of groups of cohomological dimension $2$; but this specific point does not seem to be the focus of their paper. Furthermore, theirs is not the first example of a non-finitely presented FP group, and so it leads me to my question:

Were there earlier examples of non finitely-presented FP groups of cohomological dimension $2$ ?

I want to be liberal with the meaning of the word "example": my question is specifically whether there was an earlier proof that such things existed, whether or not some explicit examples were given.

(NB : this is not quite my field of research, so maybe this is (very?) classical : any pointers to classical literature on the topic where this question is discussed would be helpful)


1 Answer 1


The Bestvina-Brady construction of non-finitely presented groups of type FP produces groups of cohomological dimension two. Bestvina-Brady groups are parametrized by finite flag simplicial complexes. The Bestvina-Brady group is of type FP iff the flag complex is acyclic (=has the same ordinary homology as a point), is finitely generated iff the flag complex is connected, is finitely presented iff the flag complex is simply connected. For an acyclic flag complex the cohomological dimension of the Bestvina-Brady group is equal to the dimension of the flag complex.
So if you apply the Bestvina-Brady construction to any non-simply connected 2-dimensional acyclic complex you get a group having the properties that you asked for.
My `generalized Bestvina-Brady groups' allow you to construct uncountable families of groups of type FP, each having cohomological dimension two. The main novelty in my article with Tom Brown that you referenced in your question is that we use very different methods, whereas previous constructions all used CAT(0) cubical techniques.
As for some references: for a discussion of the question before it was answered, look at K S Brown's book Cohomology of Groups, Chapter VIII, especially VIII.5-VIII.8. The article by Bestvina and Brady is the best place to look for the construction of the examples.

  • $\begingroup$ Thanks, this is exactly the kind of information I wanted ! $\endgroup$ Feb 14, 2022 at 14:04
  • $\begingroup$ Ian, could you explain why "For an acyclic flag complex [$L$] the cohomological dimension of the Bestvina-Brady group [$BB_L$] is equal to the dimension of the flag complex"? Clearly the dimension of the RAAG $A_L$ is at most $\mathrm{dim}L+1$, so this also bounds $\mathrm{dim}BB_L$. But how do we lose the extra 1? $\endgroup$
    – HJRW
    Feb 15, 2022 at 10:14
  • 1
    $\begingroup$ This uses the Morse function on the universal covering space of the Salvetti complex, which gets called $X_L$. The RAAG $A_L$ acts freely cocompactly on $X_L$, and $BB_L$ acts freely cocompactly on the level set (= inverse image of a point under the Morse function $f:X_L\rightarrow \mathbb{R}$). The whole of $X_L$ is built by attaching cones on $L$ to the level set. In the case when $L$ is acyclic, it follows that the level set is itself acyclic, since it must have the same homology as $X_L$ which is contractible. Thus the dimension of the level set is an upper bound for $cd(BB_L)$. $\endgroup$
    – IJL
    Feb 15, 2022 at 15:17

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