Marc Troyanov (EPFL)

Title:

Asymptotic Geometry in SOL

 

Abstract :

SOL is one of Thurston's eight classical homogeneous Riemannian geometries, possibly the most exotic one.

To get some insight of this geometry, it might be helpful to visualise the shape of a large spheres in SOL. Clearly, the first challenge is to compute, or at least estimate, the Riemannian distance between two points. In this talk, I will propose a way to circumvent this difficulty by replacing the Riemannian metric with an asymptotically equivalent Finsler metric, inspired by architectural cardboard models. This alternative Finsler metric offers the double advantage of explicit computability of distances, coupled with a rapid convergence that closely aligns with the Riemannian metric, thus simplifying our understanding and representation of SOL geometry.

As concrete applications, I will show how to represent the shape of large spheres in SOL and I will compute the volume entropy of this manifold.