Classifying incompressible surfaces in hyperbolic 4-punctured sphere mapping tori

Geometry Topology Seminar
Monday, November 25, 2019 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Sunny Yang Xiao – Brown University –
Marissa Loving

One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path satisfying certain combinatorial properties in the arc complex of the 4-punctured sphere, and how we can reconstruct such surfaces from these paths. This extends and generalizes results of Floyd, Hatcher, and Thurston.