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

TL;DR

This paper reviews the dynamics of automorphism groups on complex surfaces, introduces new classification results, and explores specific examples including folding plane pentagons and groups by Jérémy Blanc, highlighting novel dynamical behaviors.

Contribution

It provides a comprehensive overview of previous work, presents new classification results, and analyzes two novel families of examples with unique dynamical properties.

Findings

Classification of automorphism group dynamics on surfaces

Identification of new dynamical features in specific examples

Open problems in the field are discussed

Abstract

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.