Adelic version of Margulis arithmeticity theorem

Abstract.

We generalize Margulis's S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be the set of all primes including infinite one $\infty$ and set ${mathbb Q}_{\infty}={\mathbb R}$. Let S be any subset of R. For each $p\in S$, let $G_p$ be a connected semisimple adjoint ${mathbb Q}_p$-group and $D_p\subset G_p({\mathbb Q}_p)$ be a compact open subgroup for each finite prime $p\in S$. Let $(G_S, D_p)$ denote the restricted topological product of $G_p({\mathbb Q}_{p})$'s, $p\in S$ with respect to $D_p$'s. Note that if S is finite, $(G_S, D_p)=\prod_{p\in S}G_p({\mathbb Q} _{p})$. We show that if $\sum_{p\in S}\text{rank}_{ {\mathbb Q}_{p}}(G_p)\geq 2$, any irreducible lattice in $(G_S, D_p)$ is a rational lattice. We also present a criterion on the collections $G_p$ and $D_p$ for $(G_S, D_p)$ to admit an irreducible lattice. In addition, we describe discrete subgroups of $(G_{\mathbb A}, D_p)$ generated by lattices in a pair of opposite horospherical subgroups.