Remarks on discrete subgroups with full limit sets in higher rank Lie groups

We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of G = SL(3,R) must have a full limit set in the Furstenberg boundary of G. In the appendix, we show the the existence of Zariski-dense discrete subgroups Γ of SL(n,R), where n≥ 3, such that the Jordan projection of some loxodromic element γ ∈Γ lies on the boundary of the limit cone of Γ.