A weekly seminar on topics in discrete geometry with a combinatorial
We meet on Wednesdays 2:00-3:00pm in South Hall, 4607B.
more information, contact the organizers: Gordon
Dougherty, and Jon McCammond.
Subgroups of C, Knots and Tori
| April 18
In this talk, we'll discuss the following beautiful result of Hubbard & Pourezza:
The space of closed subgroups of the complex plane is homeomorphic to the four-sphere, with the collection of lattices forming an open dense subset with complement a two-sphere. Furthermore, this two-sphere is not locally-flat embedded, and intersections with three-spheres of constant latitude form trefoil knots.
We will begin by introducing the Chabauty topology on the space of closed subgroups of a topological group, and as a warm-up compute the Chabauty space of R. We'll take a detour into Seifert fibered spaces and discuss SL(2,R)/SL(2,Z) before tackling the actual problem of computing the Chabauty space of C. We will then discuss an interpretation of this space that's relevant when considering moduli spaces of tori.
This talk is meant to be elementary with the main goal of introducing the Hausdorff metric, Chabauty topology, Seifert fibrations and suspensions to younger graduate students, so first and second years are especially encouraged to stop by!