Visions in Arithmetic and Beyond: Celebrating Peter Sarnak's Work and Impact

Abstract: Discrete subgroups of PSL(2,C) are called Kleinian groups and they are fundamental groups of complete oriented hyperbolic 3-manifolds/orbifolds. Except for countably many conjugacy classes, all Kleinian groups have infinite co-volume in PSL(2, C). 

After a brief review of the Mostow-Sullivan rigidity theorem, we will discuss a new rigidity theorem for a Kleinian group and its various implications including a cross-ratio rigidity.  A Kleinian group which admits a finite sided fundamental domain without cusp is called convex cocompact. For a convex cocompact Kleinian group, we also present a measure-theoretic rigidity theorem in terms of the Hausdorff measure on the limit set.

We relate our rigidity problem with the Zariski density of the associated self-joining subgroup of the product PSL(2, C)\times PSL(2,C), which is a rank two semisimple real algebraic group. It is often hard to decide whether a given discrete subgroup of a semisimple real algebraic group is Zariski dense or not. We will discuss how we use the dynamics of diagonal flows on higher rank homogeneous spaces of infinite volume to prove these rigidity results.  

(This talk is based on joint work with Dongryul Kim)