Videos

Pathways Workshop: Topological and Geometric Structures in Low Dimensions & Geometry and Dynamics for Discrete Subgroups of Higher Rank Lie Groups: On the number of essential surfaces in cusped 3-manifolds

Presenter
January 23, 2026
SLMath
Keywords:
  • low-dimensional topology
  • geometric structures
  • homogeneous dynamics
  • Anosov representations
  • Higher rank Lie groups
  • symmetric spaces of non- compact type
MSC:
  • 57M50 - General geometric structures on low-dimensional manifolds
  • 57M60 - Group actions on manifolds and cell complexes in low dimensions
  • 57N16 - Geometric structures on manifolds of high or arbitrary dimension
  • 57S25 - Groups acting on specific manifolds
  • 22E40 - Discrete subgroups of Lie groups
  • 22F30 - Homogeneous spaces
  • 37E05 - Dynamical systems involving maps of the interval
  • 37E10 - Dynamical systems involving maps of the circle
  • 37E30 - Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces
  • 37E35 - Flows on surfaces
  • 30F60 - Teichmüller theory for Riemann surfaces
  • 32G15 - Moduli of Riemann surfaces
  • Teichmüller theory (complex-analytic aspects in several variables)]
  • 30F40 - Kleinian groups (aspects of compact Riemann surfaces and uniformization)
Play Video
Abstract
It is natural to ask how many isotopy classes of embedded essential surfaces lie in a given 3-manifold. The first bounds on the number of such surfaces were exponential, using normal surfaces. More recently, by restricting to alternating link complements in 3-sphere, Hass, Thompson and Tsvietkova obtained polynomial bounds, but for a limited class of surfaces: closed and spanning ones. In this talk, we discuss how to complete the picture for classical alternating links, and how to extend these results to other classes of cusped 3-manifolds. We give explicit polynomial bounds on all embedded essential surfaces, closed or any boundary slope, orientable or non-orientable. Our 3-manifolds are complements of links with alternating diagrams on wide classes of surfaces in broad families of 3-manifolds. This includes all alternating links in 3-sphere as well as many non-alternating ones, alternating virtual knots, many toroidally alternating knots, and most Dehn fillings of such manifolds.This is joint work with A. Tsvietkova.