Deformed relativity symmetries and the local structure of spacetime

A spacetime interpretation of deformed relativity symmetry groups was recently proposed by resorting to Finslerian geometries, seen as the outcome of a continuous limit endowed with first order corrections from the quantum gravity regime. In this work we further investigate such connection between deformed algebras and Finslerian geometries by showing that the Finsler geometries associated to the generalisation of the Poincar\'{e} group (the so called $\kappa$-Poincar\'{e} Hopf algebra) are maximally symmetric spacetimes which are also of the Berwald type: Finslerian spacetimes for which the connections are substantially Riemannian, belonging to the unique class for which the weak equivalence principle still holds. We also extend this analysis by considering a generalization of the de Sitter group (the so called $q$-de Sitter group) and showing that its associated Finslerian geometry reproduces locally the one from the $\kappa$-Poincar\'{e} group and that itself can be recast in a Berwald form in an appropriate limit.