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user6976
user6976

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 coherrencecoherence of $SL_3(\mathbb{Z})$.

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 coherrence of $SL_3(\mathbb{Z})$.

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 $SL_3(\mathbb{Z})$.

Source Link
user6976
user6976

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 coherrence of $SL_3(\mathbb{Z})$.