TITLE: Equivalent descriptions of Coherent Groups
QUESTION [8 upvotes]: 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.

REPLY [8 votes]: 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 $SL_3(\mathbb{Z})$.