Timeline for The structure of $PSL_2$ over the p-adic integers

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Dec 12, 2019 at 6:46	comment	added	ayberkz		Thanks a lot. Any book/article suggestions for learning virtually pro-$p$ and $p$-adic analytic groups? @YCor: Is there an easy way to see why $\mathrm{PSL}(2,\mathbf{Z}_p)$ doesn't split as a nontrivial free product/amalgam.
Dec 10, 2019 at 22:21	comment	added	YCor		Well $PSL(2,\mathbf{Z}_p)$ itself is $p$-adic analytic but not pro-$p$ and the characterization is among pro-$p$-groups (which is fine since a $p$-adic analytic profinite group is virtually pro-$p$).
Dec 10, 2019 at 19:32	comment	added	Ian Agol		Building on @YCor's comment, there is an abstract characterization of $p$-adic analytic groups (so the kernel $PSL(2,\mathbb{Z}_p)\to PSL(2,\mathbb{Z}/p\mathbb{Z})$) as $p$-powerful groups. en.wikipedia.org/wiki/Powerful_p-group
Dec 10, 2019 at 12:46	comment	added	YCor		It's a profinite, actually virtually pro-$p$ and $p$-adic analytic group, so reading about such topological groups might be helpful. Even as an abstract group, it doesn't split as a nontrivial free product or even amalgam.
Dec 10, 2019 at 12:15	history	edited	LeechLattice	CC BY-SA 4.0	PSL(2,Z_p) could mean PSL_2 on a finite field of p elements; edited to avoid confusion.
Dec 10, 2019 at 11:33	history	edited	ayberkz	CC BY-SA 4.0	added 22 characters in body
Dec 10, 2019 at 11:23	history	asked	ayberkz	CC BY-SA 4.0