A rigidity theorem for complex Kleinian groups

Abstract Farre, Pozzetti, and Viaggi proved that any $(d-k)$-hyperconvex subgroup of $\textsf{PSL}(d,\mathbb C)$ is virtually isomorphic to a convex cocompact Kleinian group and that its $\alpha _{k}$-critical exponent is at most 2. We show that a $(d-k)$-hyperconvex subgroup is isomorphic to a uniform lattice in $\textsf{PSL}(2,\mathbb C)$ if and only if its $\alpha _{k}$-critical exponent is exactly 2. Furthermore, we show that if a strongly irreducible $(d-k)$-hyperconvex subgroup has $\alpha _{k}$-critical exponent $2$, then it is the image of a uniform lattice in $\textsf{PSL}(2,\mathbb C)$ by an irreducible representation of $\textsf{PSL}(2,\mathbb C)$ into $\textsf{PSL}(d,\mathbb C)$.