Groups around 3-manifolds minicourse

Lectures by Nathan Dunfield at the Groups around 3-manifolds workshop at CRM during June 5-9, 2023.

Lecture 1

General background about the the topology and geometry of 3-manifolds, including the statement of the Geometrization Theorem. This lecture is intended to provide a common base of knowledge for all the mini-courses, not just this one.

• Lecture notes.
• Video recording.
• Problem sheet.
• References
  • Allen Hatcher, Notes on basic 3-manifold topology.
  • Bruno Martelli, An Introduction to Geometric Topology.
  • Aschenbrenner, Friedl, and Wilton, 3-manifold groups.

Lecture 2

Solvability of the homeomorphism problem for 3-manifolds. Practical demonstration.

• Lecture notes.
• Video recording.
• Problem sheet.
• References
  • Greg Kuperberg, Algorithmic homeomorphism of 3-manifolds as a corollary of geometrization.
  • Culler, Dunfield, Goerner, Weeks, et. al. SnapPy, a computer program for studying the geometry and topology of 3-manifolds.

Lecture 3

Finding hyperbolic structures by solving Thurston's gluing equations.

• Lecture notes.
• Video recording.
• Problem sheet.
• References
  • Jeff Weeks, Computation of Hyperbolic Structures in Knot Theory
  • Thurston's Lecture Notes, Chapters 3 and 4.
  • Jeff Weeks, Convex hulls and isometries of cusped hyperbolic 3-manifolds. Topology Appl. 52 (1993), no. 2, 127–149.

Lecture 4

Fun with finite-covers of 3-manifolds: A story of topology, geometry, and arithmetic. Basically, torsion growth in towers of finite covers a la Bergeron-Venkatesh, Lê, and Lück.

• Lecture notes.
• Video recording.
• Plots of torsion growth.
• References
  • N. Bergeron and A. Venkatesh, The asymptotic growth of torsion homology for arithmetic groups.
  • Thang Le Growth of homology torsion in finite coverings and hyperbolic volume.