Are all torsion-free finitely generated linear groups over $\mathbb{C}$ left orderable? In particular, are torsion-free congruence subgroups of $SL\_n(\mathbb{Z})$ left orderable?



Left orderable linear groups
The answer is no for congruence subgroups of $SL(n,\mathbb{Z})$ for $n \geq 3$. This is a theorem of Dave Witte-Morris; see


MR1198459 (95a:22014) 
Witte, Dave(1-MIT)
Arithmetic groups of higher Q-rank cannot act on 1-manifolds. (English summary) 
Proc. Amer. Math. Soc. 122 (1994), no. 2, 333–340. 


