Seminar

This seminar series will take place on UC Berkeley campus, in Evans Hall room 740.

The rank of a manifold is the minimal number of generators for its fundamental group. If $S$ is a surface and $f : S \to S$ is a homeomorphism, the mapping torus $M_f$ is obtained from S x [0,1] by gluing S x 1 to S x 0 via the map $(x,1) -> (f(x),0)$. It’s easy to show that rank $M_f$ <= rank S + 1.

In 2015, Souto and I showed that if S is a closed surface and f is complicated enough, then we have rank $M_f$ = rank S + 1. It turns out that the analogous result isn’t true when S has cusps, due to a possible subtle interaction between generating sets for $\pi _1 M_f $ and the cusps of $M_f$. However, if f is complicated enough, one can prove that rank M_f ≥ rank S. This is a joint project-in-progress with Dave Futer and Matt Zevenbergen.