hi cos-top
Do there exist any hyperbolic multiply connected constant-curvature compact spaces for which the fundamental domain is a cube (with flat faces, of course) for some points in the space? Or is there a proof that this is impossible?
We know that there are (at least) two such spherical spaces, that Peter Kramer calls C_2 and C_3:
Kramer09: https://ui.adsabs.harvard.edu/#abs/2009PhyS...80b5902K/
Are there any hyperbolic ones?
Cheers Boud
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Hi Boud,
although I have no firm answer to your question, I suppose that this is not the case. You do not specify what is "compact spaces", manifolds or orbifolds? In the case of orbifolds, there are tetrahedrons and these can be glued together to obtain other spaces. For example, in Physica D 92 (1996) 101, we consider the tetrahedron T8, which can be glued together to obtain a pentahedron. However, I do not know whether you can combine them to get a cube. In the case of manifolds, I think no cube is known. Although the Dehn surgery leads to many space forms, this remains as an open question. The Weeks and Thurston manifolds are constructed by hand, so even the manifold with the smallest volume is unknown.
Cheers Ralf
On Thursday 28 February 2019 18:48:36 Boud Roukema wrote:
hi cos-top
Do there exist any hyperbolic multiply connected constant-curvature compact spaces for which the fundamental domain is a cube (with flat faces, of course) for some points in the space? Or is there a proof that this is impossible?
We know that there are (at least) two such spherical spaces, that Peter Kramer calls C_2 and C_3:
Kramer09: https://ui.adsabs.harvard.edu/#abs/2009PhyS...80b5902K/
Are there any hyperbolic ones?
Cheers Boud
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hi Ralf, all,
On Mon, 4 Mar 2019, Ralf Aurich wrote:
although I have no firm answer to your question, I suppose that this is not the case. You do not specify what is "compact spaces", manifolds or orbifolds? In the case of
I meant manifolds, sorry.
orbifolds, there are tetrahedrons and these can be glued together to obtain other spaces. For example, in Physica D 92 (1996) 101, we consider the tetrahedron T8,
https://ui.adsabs.harvard.edu/#abs/1996PhyD...92..101A
which can be glued together to obtain a pentahedron. However, I do not know whether you can combine them to get a cube. In the case of manifolds, I think no cube is known. Although the Dehn surgery leads to many space forms, this remains as an open question.
Well, noone else on this list seems to know of any. :)
The Weeks and Thurston manifolds are constructed by hand, so even the manifold with the smallest volume is unknown.
Gabai, Meyerhoff & Milley (2007) - https://arxiv.org/abs/0705.4325 - claim to have proven that "the Weeks manifold is the unique smallest volume closed orientable hyperbolic 3-manifold."
I seem to remember Vincent (Blanloeil - on the cos-top list :)) being happy about the proof. This is another interesting example of a permanent ArXiv preprint with an important result.
Cheers Boud
-
Boud Roukema -
Ralf Aurich