Arithmetic groups of higher 𝑄-rank cannot act on 1-manifolds

Abstract

Let Γ \Gamma be a subgroup of finite index in SL n ( Z ) {\text {SL}_n}(\mathbb {Z}) with n ≥ 3 n \geq 3 . We show that every continuous action of Γ \Gamma on the circle S 1 {S^1} or on the real line R \mathbb {R} factors through an action of a finite quotient of Γ \Gamma . This follows from the algebraic fact that central extensions of Γ \Gamma are not right orderable. (In particular, Γ \Gamma is not right orderable.) More generally, the same results hold if Γ \Gamma is any arithmetic subgroup of any simple algebraic group G over Q \mathbb {Q} , with Q - rank ( G ) ≥ 2 \mathbb {Q} \text {-} {\text {rank}}(G) \geq 2 .