Unmarked simple length spectral rigidity for covers (Nicholas Miller)

PATCH Seminar (joint with Bryn Mawr, Haverford, Penn, and Swarthmore)

Abstract: A long studied problem in geometry is the extent to which a manifold is determined by its collection of lengths of closed geodesics. For instance, Otal showed that any negatively curved metric on a surface is determined up to isometry by its marked length spectrum. By classic work of Fricke, the similar result is true if one restricts this function only to simple closed curves.

By celebrated constructions of Vignéras and Sunada, we now know that the corresponding statement is false when one forgets the marking, that is, there exist non-isometric surfaces which have the same collections of lengths of closed geodesics. In this talk, we will explore the extent to which surfaces arising from Sunada's construction can have the same collection of lengths of simple closed curves. Along the way we will also discuss some new general results about how simple lifts of curves can determine equivalence of covers. This is joint work with Tarik Aougab, Max Lahn, and Marissa Loving.

In the morning background talk (at 11:00am in room 264), we will review some of the ingredients that go into the research talk. Topics are likely to include constructions of hyperbolic surfaces and Teichmüller space, its boundary, and maps between Teichmüller spaces in the presence of finite covers, as well as some examples of constructions of isospectral manifolds via Sunada's method.