Arithmetic groups of higher $\textbf {Q}$-rank cannot act on $1$-manifolds

Abstract:

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