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Suppose $\Gamma$ is a torsion-free lattice of $\text{PGL}_n(\mathbf{Q}_p)$ for $n\geq 3$. Then I know that by the Margulis arithmeticity theorem, $\Gamma$ must be arithmetic. My question is does $\Gamma$ need to be congruence? I know that we have the congruence subgroup property for $\text{SL}_n$, $n\geq 3$, but am I correct in understanding that it does not hold for non-simply connected groups?

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    $\begingroup$ As far as I know, the congruence subgroup property (CSP) for lattices in $SL_n({\mathbb Q}_p)$ (even for $n\geq 3$) is open. When you say "CSP for $SL_n$ is known", what is known is CSP for the split $SL_n$ over a number field. Your lattices in $SL_n$ or $PGL_n$ over $p$-adic groups are cocompact and are not covered in the known cases of CSP. As to simply connected vs adjoint, there is an old Bourbaki talk by Serre on CSP where he discusses how CSP fails for non-simply connected groups even though the congruence subgroup kernel is central. $\endgroup$ – Venkataramana Apr 27 '18 at 3:02
  • $\begingroup$ see mathoverflow.net/questions/150342/… Jim Humphrey's answer about PGL vs SL $\endgroup$ – Venkataramana Apr 27 '18 at 6:47

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