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

Abstract

Motivated by a question of M. Kapovich, we show that the ${\mathbb{Z}}^2$ subgroups of ${\mathsf{SL}}_3(\mathbb{R})$ 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 ${\mathsf{SL}}_3(\mathbb{R})$ that are not in fact Gromov-hyperbolic. By work of Oh, it also follows that a Zariski-dense discrete subgroup $\Gamma$ of ${\mathsf{SL}}_3(\mathbb{R})$ contains a regular ${\mathbb{Z}}^2$ if and only if $\Gamma$ is commensurable to a conjugate of ${\mathsf{SL}}_3(\mathbb{Z})$. In particular, a Zariski-dense regular subgroup of ${\mathsf{SL}}_3(\mathbb{R})$ contains no ${\mathbb{Z}}^2$ subgroups.

8 pages. Minor revisions; the requirement that one pass to a finite-index subgroup in the statement of Proposition 1.6 has been removed