While going through Gratzer's "General Lattice Theory", I was surprised to learn (via some exercise) that the intersection of two finitely generated subgroups is not necessarily finitely generated. Apparently, a group for which this condition holds is said to have the "Finitely-Generated Intersection Property" (FGIP). Some quick Google-ing yields some papers which have results for specific cases, but little in regard to the property in the general case.
My question is this: What can be said in the general case about the FGIP? Is there some known necessary and sufficient criteria which a group must possess for the FGIP property to hold? Or is this property too vague for consideration in the general case?
Thanks in advance!
EDIT: I think that the following questions are also natural and related to my original post. They may be equivalent variations of the same question, but I am not sure for my own part. I apologize if they are redundant.
(1) Given a group G and two specific finitely-generated subgroups, H and K, are there necessary and sufficient conditions as to whether the intersection of H and K is finitely-generated?
(2) Given an arbitrary group G, is it a decidable problem to determine whether it possesses FGIP?
(3) Are there any known counter-examples to (2), that is, a group for which the problem of determining whether the group possesses FGIP is undecidable?