Rigidity of Kleinian groups via self-joinings: measure theoretic criterion

Abstract

Let $n,m\ge 2$. Let $\Gamma <{\text{SO}}^{\circ }(n+1,1)$ be a Zariski dense convex cocompact subgroup and $\Lambda \subset {\mathbb{S}}^n$ be its limit set. Let $\rho :\Gamma \to {\text{SO}}^{\circ }(m+1,1)$ be a Zariski dense convex cocompact faithful representation and $f:\Lambda \to {\mathbb{S}}^m$ the $\rho$-boundary map. Let $${\Lambda }_f:=\bigcup \left\{C\cap \Lambda :\begin{array}{c}C\subset {\mathbb{S}}^n\text{ is a circle such that}\\ f(C\cap \Lambda )\text{ is contained in a proper sphere }\text{in }{\mathbb{S}}^m\end{array}\right\}.$$ When there exists at least one $\Lambda$-doubly stable circle in ${\mathbb{S}}^n$ (e.g., $\Omega ={\mathbb{S}}^n-\Lambda$ is disconnected), we prove the following dichotomy: $$\text{either}{\Lambda }_f=\Lambda \text{ or }{\mathcal{H}}^{\delta }({\Lambda }_f)=0,$$ where ${\mathcal{H}}^{\delta }$ is the Hausdorff measure of dimension $\delta ={\dim }_H\Lambda$. Moreover, in the former case, we have $n=m$ and $\rho$ is a conjugation by a Möbius transformation on ${\mathbb{S}}^n$. Our proof uses ergodic theory for directional diagonal flows and conformal measure theory of discrete subgroups of higher rank semisimple Lie groups, applied to the self-joining subgroup ${\Gamma }_{\rho }=(\mathrm{id}\times \rho )(\Gamma )<{\text{SO}}^{\circ }(n+1,1)\times {\text{SO}}^{\circ }(m+1,1)$. We also obtain an analogous theorem for any divergence-type subgroup.

17 pages; 2 figures; includes an added-in-proof remark on the general case; to appear in J. Topology