Departmental Colloquium
Uri Bader
Weizmann Institute
Totally geodesic submanifolds of hyperbolic manifolds and arithmeticity
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.
Friday April 2, 2021 at 3:00 PM in Zoom