Geometry of surface homeomorphism groups

• Applicants: Dr. Jonathan Bowden; Professor Dr. Sebastian Hensel
• Subject Area: Mathematics
• Term: since 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441790099

This project is concerned with topological symmetries of finite type surfaces. Such surfaces are among the most fundamental and basic objects in geometry and topology. Although surfaces are easy to describe and have been studied intensively since the early twentieth century, their homeomorphism and diffeomorphism groups show remarkable depth and complexity, and are still a rich source of interesting open problems.In recent years, geometric group theory has proved to be a valuable tool in the study of isotopy classes of homeomorphisms or diffeomorphisms (the mapping class group). The goal of this project is to transfer some of these ideas to the realm of homeomorphism groups, in order to attack some outstanding questions involving these groups.More precisely, the main focus of this project is a geometric study of the new, fine curve graph, defined by Bowden-Hensel-Webb.Isometry Classification: We aim to make connections between the dynamics of homeomorphisms acting on the surface, and the geometry of their action on the fine curve graph. This will most likely have consequences for the continuous Zimmer program, in particular relating to the question as to whether higher rank lattices can act faithfully on surfaces.Stable Commutator Length: The refined curve graph has already proved useful in constructing unbounded quasimorphisms on surface homeomorphism groups. We aim to take this idea further and start classifying which elements have positive stable commutator length.Large-scale geometry: By recent results of Mann-Rosendal, homeomorphism groups of surfaces have a well-defined large-scale geometry. Inspired by the success of curve graphs in the study of the word geometry of mapping class groups, we will similarly use the refined curve graph in order to probe the geometry of homeomorphism groups further.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity
• International Connection: United Kingdom, USA
• Cooperation Partners: Professorin Kathryn Mann, Ph.D.; Professor Richard Webb, Ph.D.