Kleinian Groups from the Sphere at Infinity and Their Self-Joinings

A Kleinian group is a discrete subgroup of the group of orientation-preserving isometries of the hyperbolic 3-space \(\mathbb {H}^3\) . As the boundary \(\partial \mathbb {H}^3\) of \(\mathbb {H}^3\) at infinity is the Riemann sphere, we refer to \(\partial \mathbb {H}^3\) as the sphere at infinity. The main purpose of this article is to discuss how deformations and rigidity of hyperbolic 3-manifolds can be understood by studying the geometry and dynamics of corresponding Kleinian groups on the sphere at infinity. We overview classical rigidity theorems by Mostow and Sullivan, and then study Patterson–Sullivan theory for Kleinian groups. We mainly focus on Sullivan’s work on conformal measures and dynamics of Kleinian groups on their limit sets in the shere at infinity with respect to conformal measures, as well as the fractal geometry of the limit sets. We finally discuss how Patterson–Sullivan theory can be extended to higher-rank settings, introducing Quint’s work on developing the higher-rank notion of critical exponents and studying higher-rank conformal measures. We especially restrict our attention to so-called self-joinings of convex cocompact Kleinian groups, which are examples of Anosov subgroups in higher rank. In this setting we present the author’s joint work with Minsky and Oh providing a higher-rank analogue of Sullivan’s identity between the critical exponent and Hausdorff dimension of the limit set of a convex cocompact Kleinian group.