Discrete subgroups of algebraic groups over local fields of positive characteristics (Q581678)

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 \(\Gamma\) \(\subset G\) is an irreducible lattice then \(\Gamma\) admits a fundamental domain which is a union of translates of Siegel domains. As a consequence the author deduces that if G has more than one simple factor, then \(\Gamma\) is finitely generated and, by a theorem due to Venkataramana, it is arithmetic. The new result here is the case of a non-cocompact \(\Gamma\). The results are stated in an apparently more general context, namely for a discrete subgroup \(\Gamma\) of G which is what is called an L-subgroup. It turns out that any L-group in G is indeed a lattice.