Is the intersection of a descending chain of finitely generated groups finitely generated

Question

Given a strictly descending chain of finitely generated groups $$ H_1 > H_2 > H_3 > \cdots,$$ is it always the case that $H = \cap_{i=1}^{\infty} H_i$ is finitely generated?

I think the answer is no, but I can't think of a counter-example.

Answer 1

Here is a more concrete example. Let $F={\rm gr}(x,y)$ be a free group of rank two and let $F'$ be the commutator subgroup of F. I think it is clear that $F'$ has infinite rank. Let $H_n$ be a subgroup generated by the set $\{x^{2^n},y^{2^n},F'\}$. Then $H_n$ has a finite index in F and is therefore a finitely generated group for any n and $F=H_0>H_1>\ldots$. However, the intersection of all $H_n$ coincides with $F'$.