Riemannian geometry of bicovariant group lattices

Abstract

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry. Despite the noncommutativity between functions and (generalized) differential forms, for the subclass of “bicovariant” group lattices considered in this work it is possible to understand central geometric objects like metric, torsion and curvature as “tensors” with (left) covariance properties. This ensures that tensor components (with respect to a basis of the space of 1-forms) transform in the familiar homogeneous way under a change of basis. There is a natural compatibility condition for a metric and a linear connection. The resulting (pseudo-) Riemannian geometry is explored in this work. It is demonstrated that the components of the metric are indeed able to properly describe properties of discrete geometries like lengths and angles. A simple geometric understanding of torsion and curvature in particular is achieved. The formalism has much in common with lattice gauge theory. For example, the Riemannian curvature is determined by parallel transport of vectors around a plaquette (which corresponds to a biangle, a triangle or a quadrangle).