3 Edited to reflect Sapir's second comment.

Let $G_0$ be a finitely generated group, and suppose there are groups $G_i$ and $K_i$ as in the following short exact sequences

$1\to K_i\to G_{i+1}\to G_i\to 1$

with $K_i$ free and nonabelian (you may assume finitely generated), and $G_i$ commutative transitive. (If $a$ is nontrivial and $b$ and $c$ both commute with $a$, then $b$ and $c$ commute.) Does it follow that $\mathrm{rank}(G_i)\to\infty$ as $i\to\infty$? Are there examples of extensions of this sort where the rank doesn't increase?

2 Changed to reflect Sapir's comment.

Let $G_0$ be a finitely generated group, and suppose there are groups $G_i$ and $K_i$ as in the following short exact sequences

$1\to K_i\to G_{i+1}\to G_i\to 1$

with $K_i$ free and nonabelian (you may assume finitely generated). Does it follow that $\mathrm{rank}(G_i)\to\infty$ as $i\to\infty$? Are there examples of extensions of this sort where the rank doesn't increase?

1

# Ranks of iterated extensions of a group by free groups.

Let $G_0$ be a finitely generated group, and suppose there are groups $G_i$ and $K_i$ as in the following short exact sequences

$1\to K_i\to G_{i+1}\to G_i\to 1$

with $K_i$ free (you may assume finitely generated). Does it follow that $\mathrm{rank}(G_i)\to\infty$ as $i\to\infty$? Are there examples of extensions of this sort where the rank doesn't increase?