Discrete subgroups of algebraic groups over local fields of positive characteristics

Abstract

It is shown in this paper that if G is the group of k-points of a semisimple algebraic group G over a local field k of positive characteristic such that all its k-simple factors are of k-rank 1 and Γ ⊂ G is a non-cocompact irreducible lattice then Γ admits a fundamental domain which is a union of translates of Siegel domains. As a consequence we deduce that if G has more than one simple factor, then Γ is finitely generated and by a theorem due to Venkataramana, it is arithmetic.