## Topology Seminar (Main Talk): Finiteness of maximal geodesic submanifolds of hyperbolic hybrids

Seminar | March 20 | 4-5 p.m. | 3 Evans Hall

Matthew Stover, Temple University

Department of Mathematics

I will explain why large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces, answering a question of Reid and (independently) McMullen for $n=3$. These are the first examples of finite-volume $n$-hyperbolic manifolds, $n >2$, for which the collection of all finite-volume totally geodesic hypersurfaces is finite but nonempty. In this talk, I will focus mostly on dimension 3, where one can even construct link complements with this property. This is joint work with David Fisher, Jean-François Lafont, and Nicholas Miller.

c_abbott@berkeley.edu