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I just asked today the following question: On groups with finite pro-$p$ completion for all primes $p$. However, I can actually simplified what I am interested in, but as it is somewhat a more general question I will leave both of them.

It is a famous open question whether every finitely generated group without polynomial growth has word growth $\succeq e^{\sqrt{n}}$. So my question is:

Are there "known" finitely generated groups, not of polynomial growth, for which the growth is not known to be $\succeq e^{\sqrt{n}}$?

That is, groups that are at least test candidates for the above open question. I am especially interested by residually finite such examples.

A result of Grigorchuk, see Theorem E.2 in https://arxiv.org/pdf/1512.07044.pdf, excludes many potential examples, e.g., residually nilpotent finitely generated groups.

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Recently, Nekrashevych gave explicit examples of infinite simple torsion groups which have in addition intermediate growth (https://arxiv.org/pdf/1601.01033.pdf).

I don't think any lower bound is known: superpolynomial growth is proved using Gromov theorem on polynomial growth (since the group is infinite and torsion it cannot have polynomial growth).

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