# arithmetic group over function fields and its fundamental domain

Let $G$ be a semi-simple algebraic group defined over a global function field $K$. Let $S$ be a finite set of places of $K$. For a place $v$ of $K$ let $K_v$ be the completion under $v$. We take $K_S=\prod_{v\in S}K_v$ and $O=\{x\in K: x \mbox{ is integral in }K_v \mbox{ if } v\not \in S \}$. It is well known that $G(O)$ is a lattice in $G(K_S)$. Are there any fundamental domain of $G(O)$ expressed in terms of Ziegel sets?

More precisely we want the following type result. If $H$ is semi-simple $\mathbb Q$-group, then it is proved in Theorem 15.5 of Borel's book "Introduction aux groupes arithmétic" that there exists a Ziegel set $F$ and a finite subset $C\subset G(Q)$ such that $FCG(\mathbb Z)=G(\mathbb R)$. To my understanding this result is not contain in the paper of Borel and Harish-Chandra 1962 on arithmetic groups.

It will be great if one can suggest some references (in English) about it.

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It is called a Siegel set (Carl Ludwig Siegel), see also mathoverflow.net/questions/124013/… –  Dietrich Burde Apr 23 at 18:25
A key player in the study of $S$-arithmetic groups over function fields has been the Bruhat-Tits building, which substitutes for the traditional symmetric space in Lie group theory.