# On groups with finite pro-$p$ completion for all primes $p$

Say that a group has Property X if its pro-$p$-completion is finite for every prime $p$. For instance, every perfect group has Property X.

Is there a finitely generated, residually finite group $G$ such that every finitely generated subgroup of $G$ is either virtually nilpotent, or has Property X (but $G$ is not virtually nilpotent)? Furthermore, is there such $G$ with intermediate growth?

We would like to have examples where not much is known about their word growth.

Motivation: Henry Bradford and I found some lower bound on the word growth for some groups. We are looking for examples of groups for which this will be useful.

Edit: I changed the condition on the subgroups to subgroups who do not have polynomial word growth to avoid the lower bound $e^{\sqrt{n}}$ that follows from a result of Grigorchuk, see Theorem E.2 in https://arxiv.org/pdf/1512.07044.pdf.

Edit: see Groups with unknown word growth for a refinement of this question.

• Such groups do not exist. Indeed the trivial subgroup and, moreover, any cyclic subgroup does not satisfy one of your conditions. – Mark Sapir Dec 14 '17 at 14:49
• @MarkSapir thanks. I have fixed it (I hope). – Yiftach Barnea Dec 14 '17 at 14:55
• To answer the question in the title: any perfect finitely generated infinite residually finite group works (since it's perfect, its pro-$p$-completion is trivial, while its profinite completion is infinite). For instance $\mathrm{SL}_3(\mathbf{Z})$ is such a group. – YCor Dec 14 '17 at 16:42
• Also beware that Bartholdi's Theorem E2 is most likely miswritten. Indeed for a f.g. group, "residually virtually nilpotent" is equivalent to "residually finite" (since virtually nilpotent f.g. groups are residually finite, and finite groups are residually nilpotent). But I don't expect its sentence starting with "In particular, if $G$ is residually virtually nilpotent... blah" to be known to hold (as it would assert a gap between polynomial and $e^{\sqrt{n}}$ for residually finite fg groups). It's probably meant to hold when $G$ is virtually residually nilpotent. – YCor Dec 14 '17 at 21:01
• ... eventually I edited the question for clarification. I think it reflects the OP's request. – YCor Dec 14 '17 at 21:11