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 $G=\mathrm{SL}(3,\mathbb{R})$ 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 $\mathrm{\Gamma }$ of $\mathrm{SL}(n,\mathbb{R})$, where $n\ge 3$, such that the Jordan projection of some loxodromic element $\gamma \in \mathrm{\Gamma }$ lies on the boundary of the limit cone of $\mathrm{\Gamma }$.