Let $G$ be an infinite finitely generated residually finite group. Is it true that $G$ contains finite-index subgroups of arbitrarily large index?
What about the converse: does there exist a finitely generated group with finite-index subgroups of arbitrarily large index that is not resitually finite? Is there any example of such a group that is furthermore amenable?
Best Answer
Nope. There's a simpler argument, not relying on finite generation, that in every infinite residually finite group the subgroups of finite index have arbitrarily large index. Let $G$ be such a group and let $g_1 \in G$ be nonzero. By residual finiteness it avoids some finite index subgroup $H_1$. Now let $g_2 \in H_1$ be nonzero. By residual finiteness it avoids some finite index subgroup $H_2$. The intersection $H_1 \cap H_2$ is also finite index and has index strictly larger than $H_1$. Now let $g_3 \in H_1 \cap H_2$ be nonzero, etc. Continuing in this way we build a sequence $H_i$ of subgroups of strictly increasing index. (And we know that each $H_i$ has a nonzero element because $G$ is infinite.)