Divergence-type semigroups in Kleinian groups and Hausdorff dimension

Let $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ is equal to the critical exponent of $\mathrm{\Gamma }$. This gives a different proof of a theorem by M. Mj and W. Yang. Along the way, we construct subsemigroups of $\mathrm{\Gamma }$ of divergence type and develop the Patterson–Sullivan theory.