Divergence-type semigroups in Kleinian groups and Hausdorff dimension
Inhyeok Choi
TL;DR
The paper investigates how the Hausdorff dimensions of Myrberg and sublinear conical limit sets for non-elementary Kleinian groups relate to the group’s critical exponent. It introduces divergence-type subsemigroups within a Kleinian group and develops a Patterson–Sullivan framework for these semigroups to study their limit sets. The main result proves that for non-convex-cocompact Γ, the dimension of the intersection Λ_Myr Γ ∩ Λ_sublinear Γ equals δ_Γ, extending previous work and enabling a semigroup-based PS analysis. The approach provides a robust method to link geometric growth, conical dynamics, and sublinear geodesic behavior in hyperbolic spaces, with potential generalizations to Gromov-hyperbolic settings. This work also clarifies the role of divergence-type phenomena in the Hopf–Tsuji–Sullivan paradigm for semigroups and Myrberg-type limit points.
Abstract
Let $Γ$ 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 $Γ$ is equal to the critical exponent of $Γ$. This gives a different proof of a theorem by M. Mj and W. Yang. Along the way, we construct subsemigroups of $Γ$ of divergence type and develop the Patterson--Sullivan theory.