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As is well known, the group $\mathrm{PSL}(2,\mathbf{Z})$ is isomorphic to the free product of two cyclic groups of orders 2 and 3.

Is there a similar description of the projective special linear group over p-adic integers? If yes, where can I find it? If no, what is known about the algebraic structure of $\mathrm{PSL}(2,\mathbf{Z}_p)$? Where to find it?

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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Dec 10, 2019 at 12:46
  • $\begingroup$ 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 $\endgroup$
    – Ian Agol
    Commented Dec 10, 2019 at 19:32
  • $\begingroup$ 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-$p$). $\endgroup$
    – YCor
    Commented Dec 10, 2019 at 22:21
  • $\begingroup$ 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. $\endgroup$
    – ayberkz
    Commented Dec 12, 2019 at 6:46

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