Quasi-isometry and left invariant orderability for groups Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a torsion-free group quasi-isometric (in the sense of Gromov) to $G$, can we conclude $H$ is left orderable? 
If the answer is no in the general case, what about 3-manifold groups?
 A: This answer adds some details, and some references, following Ian's comment.  
First the references: 
https://arxiv.org/abs/math/0211110, 
https://arxiv.org/abs/math/0203192v2
Now, for the answer: In the Boyer, Rolfsen, Wiest paper (first reference) we find:

Theorem 1.8 For each of the eight 3-dimensional geometries, there exist closed, connected, orientable 3-manifolds with the given geometric structure whose fundamental groups are left-orderable. There are also closed, connected, orientable 3-manifolds with the given geometric structure whose groups are not left-orderable.

As Ian mentions, for any pair of closed three-manifolds $M$ and $N$ with the same Thurston geometry, the fundamental groups $\pi_1(M)$ and $\pi_1(N)$ are quasi-isometric.  
In the second reference, by Calegari and Dunfield, there is a table of closed hyperbolic rational homology three-spheres, of low volume, with the orderability of the fundamental group given as orderable, non-orderable, or unknown.  It looks like this is a difficult property to determine.  Also, in this situation, orderability appears to be rare (but possible!).
A: There are also lots of amenable examples: by a theorem of P. Linnell and D. Witte-Morris (in this paper), an amenable group is left-orderable if and only if it is locally indicable (i.e. any non-trivial subgroup surjects onto $\mathbb Z$). It is not hard to see that this implies that any crystallographic group with non-solvable holonomy is not left-orderable; on the other hand any such group contains a free abelian group of finite index, and the latter is certainly orderable. One can also get examples more complicated olycyclic examples.
A simpler crystallographic example is given by the subgroup of ${\rm Isom}(\mathbb{R}^3)$ generated by three half-turns with disjoint axes of orthogonal directions (its abelianization is finite; in fact it has a presentation given by 
$$
\langle a,b | ab^2a^{-1}=b^{-2},\, ba^2b^{-1} = a^{-2}\rangle
$$
(see for example the last section in this paper of Bowditch).
