Rigidity of Kleinian groups via self-joinings

Abstract

Let $\Gamma<\mathrm{PSL}_2(\mathbb{C})\simeq \mathrm{Isom}^+(\mathbb{H}^3)$ be a finitely generated non-Fuchsian Kleinian group whose ordinary set $\Omega=\mathbb{S}^2-\Lambda$ has at least two components. Let $\rho : \Gamma \to \mathrm{PSL}_2(\mathbb{C})$ be a faithful discrete non-Fuchsian representation with boundary map $f:\Lambda\to \mathbb{S}^2$ on the limit set. In this paper, we obtain a new rigidity theorem: if $f$ is {\it conformal on $\Lambda$}, in the sense that $f$ maps every circular slice of $\Lambda$ into a circle, then $f$ extends to a M\"obius transformation $g$ on $\mathbb{S}^2$ and $\rho$ is the conjugation by $g$. Moreover, unless $\rho$ is a conjugation, 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 answers a question asked by McMullen on the rigidity of maps $\Lambda\to \mathbb{S}^2$ sending vertices of every tetrahedron of zero-volume to vertices of a tetrahedron of zero-volume. The novelty of our proof is a new viewpoint of relating the rigidity of $\Gamma$ with the higher rank dynamics of the self-joining $(\mathrm{id} \times \rho)(\Gamma)<\mathrm{PSL}_2(\mathbb{C})\times \mathrm{PSL}_2(\mathbb{C})$.