This workshop, sponsored by AIM and the NSF, will be devoted to recent developments and new directions in the study of surfaces of infinite type.
Surfaces of finite type and their mapping class groups have been the object of intensive study over the last several decades. By contrast, surfaces of infinite type and their mapping class groups, often called "big mapping class groups", are much more mysterious, and even the most basic questions about them remain open. Nevertheless, they also arise naturally: they are connected to problems in dynamics (group actions on surfaces, complex dynamics) and to the theory of taut foliations of 3-manifolds.
Recently, there has been surge of interests in surfaces of infinite type and big mapping class groups. First results have been established, but there are still many open questions. This includes for example how the algebraic invariants of big mapping class groups are related to the topological properties of the underlying surface. Another open question is to find an analogue of the Nielsen-Thurston classification for big mapping class groups, and, for some remaining cases, an analogue of the curve complex on which big mapping class groups have an interesting action. A further aspect is to study surfaces of infinite type that are equipped with a more rigid structure, such as translation surfaces. This includes the search for an analogue of the moduli space of translation surfaces and the study of the behavior of Veech groups in the infinite-type setting.
The goal of this workshop is to bring together several small but active communities working on various aspects of this young field. The main topics of the workshop are:
- Algebraic properties of big mapping class groups
- Classification problems and hyperbolic complexes
- Translation surfaces of infinite type
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
For more information email workshops@aimath.org