On Borel Anosov subgroups of $\mathrm{S}\mathrm{L}(d,\mathbb{R})$

We study the antipodal subsets of the full flag manifolds $\mathcal{F}({\mathbb{R}}^d)$. As a consequence, for natural numbers $d\ge 2$ such that $d\mathrm{\ne }5$ and $d\equiv ̸0,\pm 1\mathrm{m}\mathrm{o}\mathrm{d}8$, we show that Borel Anosov subgroups of $\mathrm{S}\mathrm{L}(d,\mathbb{R})$ are virtually isomorphic to either a free group or the fundamental group of a closed hyperbolic surface. This gives a partial answer to a question asked by Andrés Sambarino. Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space $X_d$ of $\mathrm{S}\mathrm{L}(d,\mathbb{R})$.