Geometry Seminar

The systole of a closed hyperbolic manifold is the minimal length of a nontrivial closed geodesic. The systole of such a manifold says something deep about how symmetric, and conversely how pinched, the manifold is.
Question: How does the systole grow up a tower of covers? For an arithmetic hyperbolic manifold and its covers, the systole can be analyzed using number theoretic techniques. In this talk, I will outline the history of the problem, the relevant connections between hyperbolic geometry and number theory, and then discuss recent joint work with Benjamin Linowitz and Sara Lapan in which we show that for all arithmetic hyperbolic manifolds, the systole growth up a p-congruence tower is at least logarithmic in volume. This result adds to the heuristic that, in some sense, congruence covers are the most symmetric of covers. In particular, this result can be understood as a sort of dual to the result that the Cheeger constant up a p-congruence tower is uniformly bounded from below. Speaker(s): Jeffrey Meyer (CSUSB)