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http -> https (the question has been bumped anyway)

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edited Jan 30, 2020 at 20:51

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This answer adds some details, and some references, following Ian's comment.

First the references:

http://arxiv.org/abs/math/0211110 https://arxiv.org/abs/math/0211110, http://arxiv.org/abs/math/0203192v2 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!).

This answer adds some details, and some references, following Ian's comment.

First the references:

http://arxiv.org/abs/math/0211110, http://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.

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.

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created Sep 7, 2014 at 9:26

Sam Nead

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This answer adds some details, and some references, following Ian's comment.

First the references:

http://arxiv.org/abs/math/0211110, http://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.