Uri Bader : Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity

Posted by Ian Tobasco , part of the Departmental Colloquium.

At

April 2, 2021, 3 p.m.

In

Zoom

Abstract

Compact hyperbolic manifolds are very interesting geometric objects. Maybe surprisingly, they are also interesting from an algebraic point of view: They are completely determined by their fundamental groups (this is Mostow's Theorem), which is naturally a subgroup of the rational valued invertible matrices in some dimension, GL_n(Q). When the fundamental group essentially consists of the integer points of some algebraic subgroup of GL_n we say that the manifold is arithmetic. A question arises: is there a simple geometric criterion for arithmeticity of hyperbolic manifolds? Such a criterion, relating arithmeticity to the existence of totally geodesic submanifolds, was conjectured by Reid and by McMullen. In a recent work with Fisher, Miller and Stover we proved this conjecture. Our proof is based on the theory of AREA, namely Algebraic Representation of Ergodic Actions, which Alex Furman and I have developed in recent years. In this colloquium talk I will survey the subject in a colloquial manner.