I've been reading a proof concerning S-arithmetic subgroups of algebraic groups and I'm having trouble determining why the following step should be true. First, the setup:


Let $G$ be a connected absolutely simple adjoint algebraic group defined over a nonarchimedean local field $F$. Let $K \subset F$ be a global field and let $v\_0$ be the pullback of the normalized valuation on $F$ using the embedding $K \hookrightarrow F$. Let $S$ be a set of discrete valuations on $K$ such that $S$ contains all archimedean places and does not contain any place where $G$ is anisotropic. Let $\mathcal{O}\_K(S)$ be the ring of $S$-integers in $K$. Let $\Gamma = G(\mathcal{O}\_K(S))$.


Let $\displaystyle G\_S = \prod\_{v \in S} G(K\_v)$.


We are given that $\Gamma$ is discrete in $G(K\_{v\_0})$. We know from Margulis' "Discrete Subgroups of Semisimple Lie Groups," Theorem 3.2.5, that $\Gamma$ is a lattice in $G\_S$.


The paper I'm reading concludes that $G(K\_v)$ must be compact for all $v \in S \setminus \{ v\_0 \}$. The thing that I'm having difficulty understanding is how this follow from the discreteness of $\Gamma$ and the fact that $\Gamma$ is a lattice in $G\_S$. Can someone point me in the right direction?



S-arithmetic subgroup question
This is strong approximation. $\Gamma$ is an $irreducible$ lattice in the product group $G\_S$. If there is more than one place $v$ in $S$ other than $v\_0$ where $G(K\_v)$ is not compact, then the projection of $\Gamma$ to $G(K\_{v\_0})$ is dense, and therefore cannot be discrete. Margulis' book itself may contain this result that $\Gamma$ is an irreducible lattice. Otherwise, you may look at the book by Platonov and Rapinchuk "Algebraic Groups and Number Theory" where these facts are explained and proved.


I have assumed, in using strong approximation, that $G$ is $simply \quad connected$. You are assuming that $G$ is of adjoint type. However, once you conclude that $G(K\_v)$ is compact for all other places for th simply connceted group, the same follows for the adjoint group. 


