Working Group VII:

Hyperbolic Geometry and Regge Calculus

Members: K. Mecke, C. Meusburger

Remarkably, hyperbolic geometry plays an important role both in 3 dimensional gravity and Statistical Physics. It is closely connected to the globally hyperbolic 3 dimensional Lorentzian manifolds that arise as solutions of the Einstein equations in three dimensions. In statistical physics, Riemann surfaces and tesselations of the Poincare disc arise in the modelling of condensed matter [40]. In particular, the EPINET project (see http://epinet.anu.edu.au) explores 2D hyperbolic tilings as a source of crystalline frameworks or networks in 3D euclidean space [40]. Also triply–periodic minimal surfaces are hyperbolic geometries which are widely used in the natural sciences and have extensively been studied in computational geometry. The combination of expertise from both fields will improve the understanding of the physical properties of gauge invariant observables in 3 dimensional gravity with possible implications and lessons for the 4 dimensional situation. In particular, it would be promising to combine the numerical methods developed in [40] with the results on measurements and observers in three-dimensional gravity [41] to study background radiation emerging from the initial singularity (big bang) of classical and quantised spacetimes in three dimensions.

Another fruitful encounter of both fields arises from the discretisation of 3 dimensional gravity on simplicial complexes known as Regge geometries [42]. The resulting state sum model is known as the Ponzano-Regge model and provides the simplest example of a spin foam model. Surprisingly, the treatment of the resulting statistical system with numerical methods has not been carried out yet. The lessons that one will learn by carrying out this task should provide important insight for the more complicated 4 dimensional spin foam models.

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