Tuesday, September 12, 2023
09/12/2023 - 4:15pm
For a closed hyperbolic 3-manifold M, let M_\epsilon be the set of all points in M whose injectivity radius is less than \epsilon. A lemma of Margulis implies that there is a universal constant \mu_3 such that if \epsilon < \mu_3 then M_\epsilon is topologically a disjoint union of solid tubes. This constant is useful for many finiteness arguments in hyperbolic geometry as well as effective Dehn surgery. The true value of \mu_3 is unknown, but lower bounds exist. Meyerhoff shows that \mu_3 > 0.108, while Culler, Shalen, and others have better lower bounds when the choice of M is restricted. In this talk, we will describe partial progress towards the exact value of \mu_3. In particular, we will explain how a symmetric variant of this constant can be determined and how it can be used, in preliminary work, to show that \mu_3 > 0.5. This is joint work with David Gabai and David Futer.