Dynamics of automorphism groups of projective surfaces: classification, examples and outlook

We first present an overview of our previous work on the dynamics of subgroups of automorphism groups of compact complex surfaces, together with a selection of open problems and new classification results. Then, we study two families of examples in depth: the first one comes from folding plane pentagons, and the second one is a family of groups introduced by J\'er\'emy Blanc, which exhibits interesting new dynamical features.