Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in zero energy ontology?
In zero energy ontology (ZEO) lattices in the 3-D hyperbolic manifold defined by H3 (t2-x2-y2-z2=a2) (and known as hyperbolic space to distinguish it from other hyperbolic manifolds emerge naturally. The interpretation of H3 as a cosmic time=constant slice of space-time of sub-critical Robertson-Walker cosmology (giving future light-cone of M4 at the limit of vanishing mass density) is relevant now. ZEO leads to an argument stating that once the position of the "lower" tip of causal diamond (CD) is fixed and defined as origin, the position of the "upper" tip located at H3 is quantized so that it corresponds to a point of a lattice H3/G, where G is discrete subgroup of SL(2,C) (so called Kleinian group). There is evidence for the quantization of cosmic redshifts: a possible interpretation is in terms of hyperbolic lattice structures assignable to dark matter and energy. Quantum coherence in cosmological scales could be in question. This inspires several questions. How does the crystallography in H3 relate to the standard crystallography in Eucdlidian 3-space E3? Are there general results about tesselations H3? What about hyperbolic counterparts of quasicrystals? In this article standard facts are summarized and some of these questions are briefly discussed.
For details see the article Could hyperbolic 3-manifolds and hyperbolic lattices be relevant in zero energy ontology? or the chapter TGD and Cosmology of "Physiscs n Many-Sheeted Space-time".