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Topology/Geometry Seminar

Finiteness of maximal geodesic submanifolds of hyperbolic hybrids

Matthew Stover (Temple University)

Location:  Hill 525
Date & time: Tuesday, 20 February 2018 at 3:30PM - 4:30PM

Abstract:  Reid and McMullen have both asked whether or not the presence of infinitely many finite-volume totally geodesic surfaces in a hyperbolic 3-manifold implies arithmeticity of its fundamental group. 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. 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-Fran├žois Lafont, and Nicholas Miller.

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