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Attending a series of lectures, I have recently been exposed to the notion of Coherent groups, defined as following:

Def: A group $G$ is called Coherent if every finitely generated subgroup $H$ of $G$ is finitely presented.

Examples of such groups are free, surface groups, the fundamental group of 3-manifolds. In contrast $F_2 \times F_2$ (and any group which contains it) is not coherent.

I am just wondering if this notion has any equivalent description, possibly in terms of the module category of the group algebra $kG$, or any other area.

References for further reading are highly appreciated.

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    $\begingroup$ You might be interested in the famous theorem of Bestvina--Brady, who constructed a group that's infinitely presented but of "type $FP_2$". Morally, this shows that the "obvious" module-theoretic ways of trying to recognize finite-presentedness don't work. That said, I don't think the tools exist to construct an incoherent group where every fg subgroup is of type $FP_2$. $\endgroup$
    – HJRW
    Commented Jan 18, 2018 at 11:42
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    $\begingroup$ Unlike its sheaf counterpart, I'm not aware of any use or characterization of this definition beyond what it explicitly says. $\endgroup$
    – YCor
    Commented Jan 18, 2018 at 13:46
  • $\begingroup$ @HJRW Thanks for bringing the important theorem of Bestvina-Brady to my attention, which I was not aware of. $\endgroup$
    – Kaveh
    Commented Jan 19, 2018 at 5:14
  • $\begingroup$ To elaborate on my comment, all results I'm aware on coherence directly use the definition. To prove that a group is not coherent, one exhibits a suitable subgroup. To prove that a group is coherent, one needs to have a good enough understanding of its subgroups. E.g., in the 3-manifold case, the point is that subgroups have a geometric meaning, and this relies on important topological results such that existence of a compact core. The famous case of $SL_3(\mathbf{Z})$ mentioned by Mark is intriguing because it's both hard to construct subgroups and to say anything general about its subgroups. $\endgroup$
    – YCor
    Commented Jan 19, 2018 at 16:46
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    $\begingroup$ @SamHughes No. This is problem 3 on Wise's 2020 list (and is there attributed to Serre). $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Mar 8, 2023 at 13:44

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I do not think that there is a characterization in terms of modules over the group ring. All non-trivial proofs of coherence are geometric in nature. Probably the most non-trivial (so far) is the paper by Feighn and Handel "Mapping tori of free group automorphisms are coherent" Ann. of Math. (2) 149 (1999), no. 3, 1061–1077 where it is proved that ascending HNN extensions of free groups are coherent. This implies, in particular, that almost all 1-related groups with at least 3 generators are coherent (see MR2746769). The major outstanding problem in the area is coherence of $SL_3(\mathbb{Z})$.

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  • $\begingroup$ Thanks for sharing your intuition and giving the reference to the paper by Feighn and Handel. It is interesting that their results link to the problem of coherence of groups with one relator, which I assume is still open in the general case. $\endgroup$
    – Kaveh
    Commented Jan 21, 2018 at 23:38
  • $\begingroup$ I think that the question is open for general 1-related group. It is even open for random 1-related groups with 2 generators. The statement in MR2746769 is not a simple corollary of Feighn and Handel. It uses some non-trivial properties of Brownian motions, for example. $\endgroup$
    – user6976
    Commented Jan 22, 2018 at 0:26

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