In 1979, Sullivan obtained several striking results on the asymptotic properties of a convex cocompact subgroup of $\mathrm{Isom}^{+}(\mathbb{H}^n)$, especially the identity between the critical exponent and the Hausdorff dimension of its limit set. In the recent work joint with Yair Minsky and Hee Oh, we have investigated how these can be extended to some higher-rank settings such as the product of $\mathrm{Isom}^{+}(\mathbb{H}^n)$'s.
As analogues of convex cocompact subgroups, we consider self-joinings of convex cocompact subgroups of $\mathrm{Isom}^{+}(\mathbb{H}^n)$, which are discrete subgroups of the product of $\mathrm{Isom}^{+}(\mathbb{H}^n)$'s. In this higher-rank situation, the growth indicator function and the directional limit sets are analogues of the critical exponent and the limit set, respectively.
This talk presents two main theorems. First, we provide a pointwise upper bound on the growth indicator function and discuss several interesting consequences. The second main theorem is an estimate of the Hausdorff dimension of the directional limit set of a self-joining in terms of the growth indicator function. This can be regarded as the higher-rank version of Sullivan's identity.