Non-Left-Orderability of Lattices in Higher-Rank Semisimple Lie Groups (After Deroin and Hurtado)

Let G be a connected, semisimple, real Lie group with finite centre, such that \({{\,\textrm{rank}\,}}_\mathbb {R}G \ge 2\) . B. Deroin and S. Hurtado recently proved the 30-year-old conjecture that no irreducible lattice in G has a left-invariant total order. (Equivalently, they proved that no such lattice has a nontrivial, orientation-preserving action on the real line.) We will explain many of the main ideas of the proof, by using them to prove the analogous result for lattices in p-adic semisimple groups. The p-adic case is easier, because some of the technical issues do not arise.