Extended Cartan fixed point theorem

The well-known Cartan fixed point theorem states that if a group G acts isometrically on a complete CAT(0) space and if G has a bounded orbit, then G has a fixed point. In this article, we extend the Cartan fixed point theorem to the case of gyrogroups—non-associative structures that generalize groups. We also give a prominent class of isometric gyrogroup actions on a suitable subspace of the space of bounded functions from a gyrogroup to a metric space.