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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