Timeline for Bases for free pro-p groups
Current License: CC BY-SA 3.0
7 events
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| Oct 18, 2014 at 9:35 | comment | added | Andrei Jaikin | @Pablo: Every non-trivial element of a free pro-$p$ group is a basis element of some open subgroup. So you asking whether every element of a pro-p group (the condition free is not essential) belongs to the abstract subgroup generated by a some smallest generating set. | |
| Sep 22, 2014 at 8:55 | comment | added | Pablo | @YCor: I don't think I see why there is no basis $X$ such that the abstract subgroup it generates intersects $N$. Furthermore, this question deals not with an arbitrary element but rather with a basis element of an open subgroup. | |
| Sep 22, 2014 at 8:39 | comment | added | YCor | It seems the answer is no in general. mathoverflow.net/questions/181463/…: according Barnea's answer, at least for odd $p$ there's a counterexample: namely if $N$ is the intersection of kernels of continuous homomorphisms $F\to\mathrm{GL}_2(\mathbf{Z}_p)$, then $N\neq\{1\}$ (Zubkov's Theorem) and for every basis $X$ of $F$, if $\Phi$ is the subgroup generated by $X$, we have $\Phi\cap N=\{1\}$. Zubkov's theorem is probably also true for $p=2$. | |
| Sep 21, 2014 at 21:37 | comment | added | HJRW | I wrote a paper with Zalesskii called 'Profinite properties of graph manifolds', which appeared in Geom. Dedicata. You could look at the references in that. | |
| Sep 21, 2014 at 20:01 | comment | added | Pablo | @HJRW: Thanks for this advice, I will look at this. By the way, can you give me some references (just for me to know where to start from)? | |
| Sep 21, 2014 at 19:41 | comment | added | HJRW | Pablo, I don't know how relevant it is to this specific question, but you should look into the theory of profinite and pro-p trees, as developed by Ribes and Zalesskii. (Not in their published book, but in some papers by them and various co-authors, including me.) This technology makes concrete some of the analogies between profinite (or pro-p) free groups and abstract free groups. | |
| Sep 21, 2014 at 19:07 | history | asked | Pablo | CC BY-SA 3.0 |