Embedded totally geodesic submanifolds in small volume hyperbolic manifolds- Michelle Chu (University of Illinois, Chicago)
Abstract: The smallest volume cusped hyperbolic 3-manifolds are arithmetic and contain many immersed but not embedded closed totally geodesic surfaces. In this talk we discuss nonexistence of codimension-1 closed embedded totally geodesic submanifolds in small volume hyperbolic manifolds of higher dimensions. This is joint work with Long and Reid.
Arithmetic manifolds and their geodesic submanifolds- Nicholas Miller (University of California Berkeley)
Abstract: It is a well known consequence of the Margulis dichotomy that when an arithmetic hyperbolic manifold contains one totally geodesic hypersurface, it contains infinitely many. Both Reid and McMullen have asked conversely whether the existence of infinitely many geodesic hypersurfaces implies arithmeticity of the corresponding hyperbolic manifold. In this talk, I will discuss recent results answering this question in the affirmative. In particular, I will describe how this follows from a general superrigidity style theorem for certain natural representations of fundamental groups of hyperbolic manifolds. I will also discuss a recent extension of these techniques into the complex hyperbolic setting, which requires the aforementioned superrigidity theorems as well as some theorems in incidence geometry. This is joint work with Bader, Fisher, and Stover.
Deligne-Mostow lattices and line arrangements in complex projective 2-space- Irene Pasquinelli (Institut de Mathématiques de Jussieu, Paris)
Abstract: In 1983, Hirzebruch considers some arrangements of complex lines in complex projective 2-space. Then he shows that a suitable branched cover ramified along the line arrangement is a complex hyperbolic manifold. This manifold turns out to be one of the well known Deligne-Mostow lattices. In the first part of this talk I will introduce you to the complex hyperbolic space, to its group of isometries and to the Deligne-Mostow lattices. Then I will tell you about Hirebruch's construction. In the second part, I will first explain how Hirzebruch's work has been generalised by Bartel, himself and Hoefer to all of the Deligne-Mostow lattices. Then I will tell you how, in a joint work with Elisha Falbel, we created an explicit dictionary between line arrangements and fundamental domains for the lattices. I will also explain one of the applications of this. In fact, we use this result to contribute to the problem of creating a complex analogue to the hybridisation construction.