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

Current License: CC BY-SA 3.0

15 events

when toggle format	what		by	license	comment
Dec 14, 2017 at 22:39	comment	added	Yiftach Barnea		@YCor thanks a lot. It was a bit tricky to formulate this question.
Dec 14, 2017 at 21:11	comment	added	YCor		... eventually I edited the question for clarification. I think it reflects the OP's request.
Dec 14, 2017 at 21:10	history	edited	YCor	CC BY-SA 3.0	clarified the question; edited tags
Dec 14, 2017 at 21:01	comment	added	YCor		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.
Dec 14, 2017 at 20:53	comment	added	YCor		Yiftach, there are several problems making your question unclear, which you didn't fix; please clarify: (1) the title is confusing since it suggest another easier question (2) you're talking of subgroups of $p$-power index, but most likely you mean normal subgroups of $p$-power index.
Dec 14, 2017 at 20:11	history	edited	Yiftach Barnea	CC BY-SA 3.0	added 127 characters in body
Dec 14, 2017 at 19:06	history	edited	Yiftach Barnea	CC BY-SA 3.0	added 251 characters in body
Dec 14, 2017 at 18:33	comment	added	Yiftach Barnea		The point is that if the group contains, for example, a non abelian free groups, then its word growth is already exponential.
Dec 14, 2017 at 16:47	comment	added	YCor		I would have been reasonable to ask the question in the title, and to require in addition that it holds for all subgroups of finite index (to discard examples such as $\mathrm{SL}_d(\mathbf{Z})$. Requiring something about all non-cyclic subgroups strongly narrows the scope (you forbid free subgroups, etc).
Dec 14, 2017 at 16:44	comment	added	YCor		This is distinct to the question in the text (at this very time), since it refers to subgroups of $p$-power index. The core of a subgroup of $p$-power index is often far from being of $p$-power index. For instance, the product $\prod_n\mathrm{Alt}_{p^n}$ admits subgroups of arbitrary large $p$-power index, and hence so do its dense subgroups.
Dec 14, 2017 at 16:42	comment	added	YCor		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.
Dec 14, 2017 at 14:55	comment	added	Yiftach Barnea		@MarkSapir thanks. I have fixed it (I hope).
Dec 14, 2017 at 14:53	history	edited	Yiftach Barnea	CC BY-SA 3.0	added 9 characters in body
Dec 14, 2017 at 14:49	comment	added	user6976		Such groups do not exist. Indeed the trivial subgroup and, moreover, any cyclic subgroup does not satisfy one of your conditions.
Dec 14, 2017 at 14:22	history	asked	Yiftach Barnea	CC BY-SA 3.0