Videos

Recent Progress in Topological and Geometric Structures in Low Dimensions: Towards the Nielsen-Thurston classification for surfaces of infinite type

Presenter
March 23, 2026
Keywords:
  • Surfaces
  • geometric 3-manifolds
  • mapping class groups
  • foliations
  • fibrations
  • Anosov flows
  • Teichmuller space
  • conformal structures
MSC:
  • 57M50 - General geometric structures on low-dimensional manifolds
Abstract
The fundamental theorem of Thurston states that any homeomorphism of a surface of finite type can be isotoped so that some multi-curve is invariant and so that for every complementary component the first return map is either periodic or pseudo-Anosov. Homeomorphisms of infinite type surfaces are much more complicated. In this work we focus on the class of tempered homeomorphisms -- these are the ones that do not have any mixing behavior. We show that up to isotopy there is an invariant geodesic lamination so that the first return maps display one of three qualitatively different behaviors. This work is in progress and it is joint with Federica Fanoni and Jing Tao.