Rigidity of Kleinian groups via self-joinings

Let $\mathrm{\Gamma }<{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})\simeq {\mathrm{I}\mathrm{s}\mathrm{o}\mathrm{m}}^{+}({\mathbb{H}}^3)$ be a finitely generated non-Fuchsian Kleinian group whose ordinary set $\mathrm{\Omega }={\mathbb{S}}^2-\mathrm{\Lambda }$ has at least two components. Let $\rho :\mathrm{\Gamma }\to {\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ be a faithful discrete non-Fuchsian representation with boundary map $f:\mathrm{\Lambda }\to {\mathbb{S}}^2$ on the limit set. In this paper, we obtain a new rigidity theorem: if $f$ is {\it conformal on $\mathrm{\Lambda }$}, in the sense that $f$ maps every circular slice of $\mathrm{\Lambda }$ into a circle, then $f$ extends to a Möbius 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 $f(C\cap \mathrm{\Lambda })$ is contained in a circle has empty interior in the space of all circles meeting $\mathrm{\Lambda }$. This answers a question asked by McMullen on the rigidity of maps $\mathrm{\Lambda }\to {\mathbb{S}}^2$ 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 $\mathrm{\Gamma }$ with the higher rank dynamics of the self-joining $(\mathrm{i}\mathrm{d}\times \rho )(\mathrm{\Gamma })<{\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})\times {\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$.