Hausdorff Dimension of non-conical and Myrberg limit sets

Abstract

In this paper, we develop techniques to study the Hausdorff dimensions of non-conical and Myrberg limit sets for groups acting on negatively curved spaces. We establish maximality of the Hausdorff dimension of the non-conical limit set of $G$ in the following cases. 1. $M$ is a finite volume complete Riemannian manifold of pinched negative curvature and $G$ is an infinite normal subgroups of infinite index in $\pi_1(M)$. 2. $G$ acts on a regular tree $X$ with $X/G$ infinite and amenable (dimension 1). 3. $G$ acts on the hyperbolic plane $\mathbb H^2$ such that $\mathbb H^2/G$ has Cheeger constant zero (dimension 2). 4. $G$ is a finitely generated geometrically infinite Kleinian group (dimension 3). We also show that the Hausdorff dimension of the Myrberg limit set is the same as the critical exponent, confirming a conjecture of Falk-Matsuzaki.