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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?

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Is there an infinitely generated residually finite group with a bound on the index of its finite-index subgroups?

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.)

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For a finitely generated group there are finitely many morphisms onto any given finite group. It follows that if a finitely generated group has a bound on the index of its normal subgroups, then it has finitely many subgroups of finite index, so their intersection is again finite. If this group is infinite, then the intersection of all normal subgroups of finite index is not trivial, so the group si not residually finite.

Here is an example as above: it is $ (\bigoplus\limits_{i \in \mathbb{Z}} A_5) \rtimes \mathbb{Z}$, which is amenable, finitely generated, non-residually finite and contains subgroups of arbitrarily large finite index. I can provide the details if someone is interested.

Now the question I am wondering is: is there an infinitely generated residually finite group with a bound on the index of its finite-index subgroups?

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    $\begingroup$ is your action for semidirect product same as wreath(That is what I am guessing). $\endgroup$ – Paul Plummer Dec 3 '18 at 19:07

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