In the case of the group $SO(n,1)$ there is a criterion known for whether or not two given elements of the group generate a cocompact lattice. Is any similar criterion known in the case of $PL(2,K)$ for $K$ a non-archimedean local field, or for some other $p$-adic group?
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5$\begingroup$ what is this criterion for $SO(n,1)$ please? $\endgroup$– VenkataramanaCommented Nov 5, 2015 at 10:24
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1$\begingroup$ I second the question! $\endgroup$– Igor RivinCommented Nov 5, 2015 at 12:26
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1$\begingroup$ I do not believe there is a criterion for $SO(n,1)$. There is a couple of semi-algorithms (for an arbitrary finite collection of elements of $SO(n,1)$). One of the two semi-algorithms is due to Troel Jorgensen. The second is a bit more complicated and is not in the literature; unlike Jorgensen's semi-algorithm, it applies to all real rank 1 noncompact simple Lie groups. $\endgroup$– MishaCommented Nov 5, 2015 at 18:30
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$\begingroup$ Yes, I apologise, I think I was mistaken. I was told that there was a criterion like this in Beardon's book "The Geometry of Discrete Groups" but I am having difficulty locating any result like that in the book. $\endgroup$– RupertCommented Nov 6, 2015 at 9:16
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$\begingroup$ I would like to know how to prove that if $\gamma$ is an element of a torsion-free cocompact lattice in $PL(2,K)$ for $K$ a non-archimedean local field then the eigenvalues of $\gamma$ are in $K$. Ihara appears to make this claim in his article "Discrete subgroups of $PL(2,k_{\mathcal{P}})$. $\endgroup$– RupertCommented Nov 6, 2015 at 9:49
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