# Finitely generated subgroups of infinite index

It is well known that if $G$ is a finitely presented subgroup then all its finite index subgroups are finitely presented also. I was wondering if it is possible to extend this,

What classes $\mathcal{C}$ of finitely generated/presented groups are there such if $G \in \mathcal{C}$ then every subgroup of $G$ is finitely generated/presented?

This is, in some ways, the opposite to what happens in finitely generated free groups, where a normal subgroup is of finite index if and only if it is finitely generated.

Examples of such classes of groups are the Tarski monster groups, as well as finitely generated abelian groups. I was therefore wondering if there were more interesting examples (one could argue that you cannot get more interesting that Tarski monster groups...but anyway), such as amenable, or hyperbolic, with added conditions. This is (conceivably) not too much to ask - f.g. abelian groups are amenable, while the Tarski monster groups were constructed using a sort of layered small cancellation theory (so if they aren't hyperbolic, some of them at least "look like" they are.)

• In case you don't know about this: There is the intermediate notion of coherent groups that is quite well-studied. These are the finitely presented groups with the property that every finitely generated subgroup is finitely presented. Free groups have that property (vacuously) as well as finitely generated nilpotent groups and the class of coherent groups is stable under free products, so you can build quite non-trivial examples of those.
– t.b.
Commented Sep 5, 2011 at 11:04
• Thanks. I did know about them, but had forgotten. Commented Sep 5, 2011 at 11:10
• The groups $\mathbb{Z}^n$ for $2 \leq n < \infty$ give examples of what you want. One could also say that these are not so interesting from a group theoretic perspective...but then I would encourage you to refine your question to get at what you really want. Commented Sep 5, 2011 at 13:44
• I've edited my question to be slightly more general, and would make it a community wiki if I knew how... Commented Sep 5, 2011 at 13:53
• (Virtually) Polycyclic groups fit the bill.
– user641
Commented Sep 5, 2011 at 16:24

## 1 Answer

The word I was looking for was coherent. I was basically wondering what groups are coherent.