Automorphisms of Surfaces

Abstract: The classification of compact topological surfaces goes back to the mid 19th century, in lieu of attempts to find counterexamples to Euler’s formulae for polyhedra, and is a landmark result in classical geometric topology. On the other hand the group of automorphisms of a surface is a very rich object and its study is still relevant today. The mapping class group of a surface, consisting of automorphisms up to deformation, appears naturally in many contexts Teichmüller theory, Thurston’s Geometrisation of 3-manifolds, stable homotopy theory (Madsen-Weiss’ proof of the Mumford Conjecture) to name a few. By contrast, the study of the full group of automorphisms, which is relevant in dynamics, is somewhat less understood and several natural questions remained open until recently. I will put these problems into context and indicate some recent progress concerning the geometry of this group in joint work with S. Hensel and R. Webb.