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It is known, from the works of G.Margulis, etc. that lattices in semi-simple real (algebraic) groups are "often" arithmetic subgroups, as long as the split rank is high enough. Here by a lattice in a Lie group $G$ is understood a discrete subgroup $\Gamma$ in $G$ such that the measure on $\Gamma\backslash G$ induced from the Haar measure on $G$ is of finite volume.

Typical example: for $G$ a simple $\mathbb{Q}$-group, with $\mathbb{Q}$-rank at least 2, then a lattice in $G(\mathbb{R})$ is arithmetic, in the sense that it is commensurable with a congruence subgroup, the latter being the intersection $G(\mathbb{Q})\cap K$ inside $G(\mathbb{A}^f_\mathbb{Q})$ for some compact open subgroup $K$ in the group of adelic points of $G$.

And what should be said about discrete subgroups in $G(\mathbb{Q}_p)$? Write for simplicity $G_p=G(\mathbb{Q}_p)$. At least one knows that for a discrete subgroup $\Gamma$ in $G_p$, the quotient $\Gamma\backslash G_p$ is compact if and only if it is of finite volume with respect to the Haar-induced measure. If $G$ is defined over a global field, say $\mathbb{Q}$, then a conguence subgroup of $G$ "should" be also a lattice in $G_p$. Moreover, for general semi-simple $\mathbb{Q}_p$-group $G$, how to classify the co-compact discrete subgroups in $G_p$? Do they admit explicit constructions like those coming from global fields?

Any comments and references are welcome.


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up vote 6 down vote accepted

The arithmeticity theorem of Margulis also characterizes lattices in product of simple groups over $R$ as well as $p$-adic fields. All such lattices in the higher rank case are S-arithmetic. When all but one of the factors is compact, the projection onto the non-compact factor (that you can take to be a p-adic field) gives you a lattice there. For instance, take a quadratic form $q$ that is anisotropic over $R$, but isotropic over ${\mathbb Q}_p$ for some $p$. Then $SO({\mathbb Z}[1/p])$ is a lattic in the product $SO(q, {\mathbb R}) \times SO(q, {\mathbb Q}_p)$. The projection onto the first component is dense, and onto the second component is a lattice there. For the precise statement of the (S-)arithmeticity theorem, the best place to look is Margulis's book "Discrete Subgroups of Semisimple Lie Group".

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