Non-left-orderability of lattices in higher-rank semisimple Lie groups (after Deroin and Hurtado)

TL;DR

This paper discusses the non-left-orderability of lattices in higher-rank semisimple Lie groups, extending recent proofs to p-adic cases and explaining key ideas behind these results.

Contribution

It provides a detailed explanation of the proof that lattices in higher-rank semisimple Lie groups are not left-orderable, including new results for p-adic groups.

Findings

Lattices in higher-rank semisimple Lie groups are not left-orderable.

The proof techniques for real Lie groups can be adapted to p-adic groups.

The p-adic case is technically simpler, aiding the understanding of the main ideas.

Abstract

Let $G$ be a connected, semisimple, real Lie group with finite centre, with real rank at least two. 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.