A discrete group of conformal automorphisms of the Riemann sphere is called a Kleinian group. Anosov Kleinian surface groups are quasi-conformal deformations of Fuchsian surface groups; they preserve a Jordan curve cutting the sphere into two disks on which the action of the group is nice. Bers proved that quasi-Fuchsian surface group representations are determined uniquely by the pair of conformal structures at infinity, giving a natural parameterization by a product of Teichmüller spaces. We study the higher rank analogue: hyperconvex Anosov surface group representations into $\mathrm{PSL}(d,C)$, introduced by Pozzetti—Sambarino—Wienhard. We define a natural map into a product of Teichmüller spaces of Riemann surface foliations and prove the following analogue of a famous theorem of Bowen from the 70’s: The Hausdorff dimension of the limit set of a fully hyperconvex surface subgroup into $\mathrm{PSL}(d,C)$ is equal to 1 iff it is conjugated into $\mathrm{PSL}(d,R)$. This is joint work with Beatrice Pozzetti and Gabriele Viaggi.