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.
Thanks!
