[Cos-top] hyperbolic 3D spaces with cubical FDs?

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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