The non-cocompact rank-three lattice action conjecture

Let $\Gamma$ be a non-cocompact lattice in either $\mathop{\mathrm{SL}}\nolimits(3,\mathbb{R})$ or $\mathop{\mathrm{SL}}\nolimits(3,\mathbb{C})$. Non-cocompact rank-three conjecture. $\Gamma$ has no nontrivial orientation-preserving action on $\mathbb{R}$. The source presents this as a case that its methods seem likely to prove, rather than as an established result; its status is therefore open.