Title: Ranks of cusped mapping tori Abstract: The rank of a manifold is the minimal number of generators for its fundamental group. If S is a surface and f : S -> 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.

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Last modified: Mon Apr 27 17:23:39 EDT 2026