Abstract: The Modular surface is the quotient of the hyperbolic space by the modular group $PSL_2(Z)$. Each closed geodesic on the surface corresponds naturally to a real quadratic field, and is also a closed orbit for the geodesic flow naturally defined on the unit tangent bundle of the surface. When taking a closed geodesic, or a finite union of geodesics out of the unit tangent bundle, one always gets a hyperbolic three dimensional manifold. In the talk I will show that for an infinite collection of such “modular links” the complements are in fact arithmetic three manifold, corresponding to a complex quadratic field.
This is joint work with Jose Andres Rodriguez-Migueles and Jessica Purcell.