Divergence-type semigroups in Kleinian groups and Hausdorff dimension

Abstract

Let $\Gamma$ be a non-elementary, non-convex-cocompact Kleinian group acting on $\mathbb{H}^{d}$. We show that the Hausdorff dimension of the sublinearly conical Myrberg limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$. This gives a different proof of a theorem by M. Mj and W. Yang. Along the way, we construct subsemigroups of $\Gamma$ of divergence type and develop the Patterson--Sullivan theory.