Are all torsionfree finitely generated linear groups over $\mathbb{C}$ left orderable? In particular, are torsionfree congruence subgroups of $SL_n(\mathbb{Z})$ left orderable?
The answer is no for congruence subgroups of $SL(n,\mathbb{Z})$ for $n \geq 3$. This is a theorem of Dave WitteMorris; see MR1198459 (95a:22014) Witte, Dave(1MIT) Arithmetic groups of higher Qrank cannot act on 1manifolds. (English summary) Proc. Amer. Math. Soc. 122 (1994), no. 2, 333–340. 

