On regular subgroups of ${\mathsf{S}\mathsf{L}}_3(\mathbb{R})$

Motivated by a question of M. Kapovich, we show that the ${\mathbb{Z}}^2$ subgroups of ${\mathsf{S}\mathsf{L}}_3(\mathbb{R})$ that are regular in the language of Kapovich–Leeb–Porti, or divergent in the sense of Guichard–Wienhard, are precisely the lattices in minimal horospherical subgroups. This rules out any relative Anosov subgroups of ${\mathsf{S}\mathsf{L}}_3(\mathbb{R})$ that are not in fact Gromov-hyperbolic. By work of Oh, it also follows that a Zariski-dense discrete subgroup $\mathrm{\Gamma }$ of ${\mathsf{S}\mathsf{L}}_3(\mathbb{R})$ contains a regular ${\mathbb{Z}}^2$ if and only if $\mathrm{\Gamma }$ is commensurable to a conjugate of ${\mathsf{S}\mathsf{L}}_3(\mathbb{Z})$. In particular, a Zariski-dense regular subgroup of ${\mathsf{S}\mathsf{L}}_3(\mathbb{R})$ contains no ${\mathbb{Z}}^2$ subgroups.