## Could Hyperbolic 3-Manifolds and Hyperbolic Lattices be Relevant in Zero Energy Ontology?

**Authors:** Matti Pitkanen

In zero energy ontology (ZEO) lattices in the 3-D hyperbolic manifold defined by H^{3} (t^{2}-x^{2}-y^{2}-z^{2}=a^{2}) (and known as hyperbolic space to distinguish it from other hyperbolic manifolds emerge naturally. The interpretation of H^{3} as a cosmic time=constant slice of space-time of sub-critical Robertson-Walker cosmology (giving future light-cone of M^4 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 H^{3} is quantized so that it corresponds to a point of a lattice H^{3}/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 would be in question. This inspires several questions. How does the crystallography in H^{3} relate to the standard crystallography in Eucdlidian 3-space E^{3}? Are there general results about tesselations H^{3}? What about hyperbolic counterparts of quasicrystals? In this article standard facts are summarized and some of these questions are briefly discussed.

**Comments:** 9 Pages.

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### Submission history

[v1] 2012-11-13 02:01:26

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