Rigidity of Kleinian groups via self-joinings

Group Actions, Geometry and Dynamics
Event time: 
Monday, September 12, 2022 - 4:00pm
LOM 206
Hee Oh
Speaker affiliation: 
Event description: 

Let $\Gamma<\rm{PSL}_2(\mathbb C)$ be a finitely generated non-Fuchsian Kleinian group and let $\rho$  be a faithful discrete type-preserving non-Fuchsian representation of $\Gamma$ into $\rm{PSL}_2(\mathbb C)$. Mostow’s rigidity theorem says that if $\Gamma<\rm{PSL}_2(\mathbb C)$ is a lattice then $\rho$ is trivial i.e., it is  a conjugation by a Mobius transformation.  General finitely generated Kleinian groups can be classified into two kinds by the Ahlfors measure conjecture  (which is now a theorem):  the limit set  $\Lambda$ is either the whole Riemann sphere ${\mathbb S}^2$ or has measure zero.  Sullivan’s rigidity theorem applies only to the former case:  if $\rho$ is a quasi-conformal deformation of $\Gamma$,  it is trivial. We will present a  new rigidity theorem which applies to  the latter case:   Suppose that the ordinary set $\Omega={\mathbb S}^2-\Lambda$  has at least two components. If the $\rho$-boundary map $f:\Lambda\to {\mathbb S}^2$ maps every circular slice of $\Lambda$ to a circle, then $\rho$ is trivial. Moreover, unless $\rho$ is trivial, the set of circles $C$ such that $f(C\cap \Lambda)$ is contained in a circle has empty interior in the space of all circles meeting $\Lambda$. This also answers a question of McMullen on the rigidity of $f$ mapping vertices of every zero-volume tetraheron to vertices of a zero-volume tetrahedron.

The novelty of our proof is a new viewpoint of relating the rigidity of $\Gamma$ with the higher rank dynamics of the self-joining $(id \times\rho)(\Gamma)<\rm{PSL}_2(\mathbb C)\times \rm{PSL}_2(\mathbb C)$.

This talk is based on joint work with Dongryul Kim.