Department of Mathematics

A finitely generated group G admits a natural word metric that has been popularized by Gromov. A finitely generated subgroup H<G has its own intrinsic word metric, as well as the metric induced by G. Distortion is a function measuring the degree of difference between these two metrics. In the setting of compact 3-manifolds and their surface subgroups, distortion is often closely connected with other geometric or topological properties. I will discuss one such connection between distortion and virtual embedding. An immersed surface in a 3-manifold M is virtually embedded if the immersion lifts to an embedding in a finite cover of M. In the setting of 3-dimensional graph manifolds and horizontal surfaces, we show that a surface has quadratic distortion if it is virtually embedded and has exponential distortion if it is not virtually embedded. This is joint work with Hoang Thanh Nguyen. Tea at 3:33 pm in Stevenson 1425. (Contact Person: Matthew Haulmark)