Margulis ergodicity–lattice conjecture for special linear groups

Determine whether every discrete subgroup of $SL_n(\mathbb{R})$, for $n>2$, that acts ergodically on its Furstenberg boundary must be a lattice.

Background

The paper studies the conjecture that, for a simple real Lie group of real rank at least two, ergodicity of a discrete subgroup on the Furstenberg boundary forces the subgroup to have finite covolume. The authors disprove this conjecture for $SO(n,2)$ with $n\geq 3$ by constructing infinite-covolume, including Zariski-dense, examples acting ergodically on the boundary.

The paper notes that the mechanism producing these counterexamples relies on non-tempered rank-one subgroups contained in $SO(n,2)$. Since groups such as $SL_n(\mathbb{R})$ for $n>2$ do not contain such non-tempered rank-one semisimple subgroups, it remains unresolved whether the conjecture holds in that setting.