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?



Quasi-isometry and left invariant orderability for groups
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:



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> 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.
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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!).


