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.